hpm 與高中幾何教學：以圓錐曲線的正焦弦為例
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19101985
1998 10 5 5
http://math.ntnu.edu.tw/horng
3 3
1.
Girolamo Cardano, 1501~1576 1545 Ars magna
On the Rules of Algebra
F. Viète, 1540~1603
In Artem Analyticem Isagoge, Introduction to Analytic Art, 1591
Introduction to Analytic Art
— Pappus
analysissynthesis
zetetics (seekink the truth
)
poristics
rhetics exegetics
HPM
sin( )y a bx c d= + +
a b c R A B C = = =
A B C = = 2
ABC a, b, cA(0, 0)B(c, 0) C
( cos , sin )C b A b A
2 2 2( cos ) ( sin 0)BC b A c b A= − + −
2 2 2 2 2cos 2 cos sinb A bc A c b− + + A 2 2 2 cosb c bc A+ −
BC a= 2 2 2 2 cosa b c bc= + − A
x
+ =
− =
2
(2006), HPM
/
x
a 2a x, y
33 2ax =
a y
y x
x a
y x
y = 32xy a= y
(2)
F. van Schooten (1615-1660)
HPM
AB BE B
BE D BDAB D
L E
D L E
E
θ
(a) A LK x E
A E(x, y)ABBD=a DE=b BE
θ E
x
y
A
B
D
E
(b) abθ BDE (c) (c) E E (d) van Schooten
(3)
3 van SchootenVan Maanen “Alluvial Deposits, Conic Sections, and Improper Glasses, or History of Mathematics Applied in the Classroom”, Learn From The Masters!
HPM
A. Quetelet G. Dandelin 19
F1 S2EF2S1
k1k2E1E2 PF1PE1S1 PF1PE1 PF2PE2S2PF2PE2
PF1PF2PE1+PE2E1E2
E1E2S1S2
(Conics)
(Apollonius of Perga, 262 B. C. ~ 190 B. C.)
(diameters)
111213
K KL HF, FL
K y KLx
H
L
K
GF
y
x
p
HPM
LF pHF (parameter)
(latus rectum) parabola
2y p= x
FX FL FN
OLPX F
HF x FXxMNy
FLpHFd HFFLOLLP
dpxLP LP p x d
= +
p=FL hyperbola
x
p
xd
y
O
XP
L
L LM MO
MO EM EH ON
E ED x EMx LMy EH
pEDdEDEHOXOH dpx
OH OH
2 py px x d
= − ⋅ x 2 py px x d
= − 2 pEH
ellipse
parabola., hyperbola, ellipse
(fall short) (exceed) (fit) elleipsis, “defect”hyperbole, “excess”parabole, “a placing beside” parameter
11 FH 12 FL 13 EH “the straight lines drawn ordinatewise to the diameter are applied in square”
upright side latus rectum
HPM
a : b :: c: d
:: analogia M. Fried
(figure) “upright side”
2
= ±
(the straight line cut off by it on the diameter beginning from the section’s 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.) ABC
ABC [I. 3]DE
ACFFHFG
BC : BA, AC:: FH : FA
K K KL DE KL HF, FL
HPM
(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) ABC
ABC [I. 3]DE
HAKAFGBCKFFG
FL KA : BK, KC : : FH : FL M M MN DE N NOX FL HL X LOXP FN MN FX (applied to) FL FN LX HF, FL 13
(subcontrariwise)
(the parameter)
(And let such a section be call an ellipse)
HPM
(subcontrariwise)DE
FGBCED [I. 7 and Def. 4]E
EHEDAAKED AKBK, KC : : DEEH L L LM FG LM (applied to) EH EM
DE, EH
Apollonius (1952). Conics (tr. R. C. Taliaferro), in Great Books of the Western World, Encyclopaedia Britannica.
The Philosophical Works of Descartes, translated by E. S. Haldane and G. R. T. Ross (1968). London: Cambridge at The University Press.
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”, Science in Context 16(3).
Katz, Victor. J., (1993). A History of Mathematics: An Introduction. New York: HarperCollins College Publishers.
Grattan-Guinness, Ivor, (1997). The Fontana History of the Mathematical Sciences. London:HarperCollins College Publishers.
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).
HPM
1623 6 19 (Clermont-Ferrand)1662 8 19
(Etienne Pascal, 1588-1651)
(Mersenne)
(Torricelli Evangelista,1608-1647)
(Pierre de Fermat, 1601-1665)
(Euclid) (The Elements)
1654 11
HPM
(Christian Huygens,1629 - 1695)
(Chevalier De Mere, 1607-1684)
< )……
n (x+y)
(Traite du triangle arithmetique) 1665
(Omar Khayyam, 1048-1131)
n
1654
(Isaac Newton, 1643-1727)
n ( )nx+1 x
(Pappus)
A” B” C” 2.
1646 1651 1654
1654
1658 (Traite General de la Roulette) 1655
(Pensees)
1. PDF e-mail suhui_yu@yahoo.com.tw
2. 3. e-mail
suhui_yu@yahoo.com.tw 4. http://math.ntnu.edu.tw/horng/letter/hpmletter.htm 5.
HPM ( Boston Consulting Group)
()
()
()
()
1910 11 12 1985 6 12 4
50
1900
1952
189319691896197319001979
Jacques Hadamard, 18651963Norbert Wiener, 18941964
1933
1935 1936
19112004
19101970 1939 1941
20
1946 2 5 9 1926
1924
HPM
1947 1950 3 16
(189219781952 7
19301927
1950193319961927
1983 10
California Institute of Technology
1984 4
1985 6 3 6 12 4
F.Shiratori
HPM
Critical path method, CPMProgram evaluation and review Technique, PERT
1964 18931976
3 18
——
……
—
HPM
2008
8 1999
30
2001
HPM
20
Jean-Claude Martzloff Owen Gingerich
Benno van Dalen
Karine Chemla
Bernd Zimmermann
211
suhui_yu@yahoo.com.tw
1998 10 5 5
http://math.ntnu.edu.tw/horng
3 3
1.
Girolamo Cardano, 1501~1576 1545 Ars magna
On the Rules of Algebra
F. Viète, 1540~1603
In Artem Analyticem Isagoge, Introduction to Analytic Art, 1591
Introduction to Analytic Art
— Pappus
analysissynthesis
zetetics (seekink the truth
)
poristics
rhetics exegetics
HPM
sin( )y a bx c d= + +
a b c R A B C = = =
A B C = = 2
ABC a, b, cA(0, 0)B(c, 0) C
( cos , sin )C b A b A
2 2 2( cos ) ( sin 0)BC b A c b A= − + −
2 2 2 2 2cos 2 cos sinb A bc A c b− + + A 2 2 2 cosb c bc A+ −
BC a= 2 2 2 2 cosa b c bc= + − A
x
+ =
− =
2
(2006), HPM
/
x
a 2a x, y
33 2ax =
a y
y x
x a
y x
y = 32xy a= y
(2)
F. van Schooten (1615-1660)
HPM
AB BE B
BE D BDAB D
L E
D L E
E
θ
(a) A LK x E
A E(x, y)ABBD=a DE=b BE
θ E
x
y
A
B
D
E
(b) abθ BDE (c) (c) E E (d) van Schooten
(3)
3 van SchootenVan Maanen “Alluvial Deposits, Conic Sections, and Improper Glasses, or History of Mathematics Applied in the Classroom”, Learn From The Masters!
HPM
A. Quetelet G. Dandelin 19
F1 S2EF2S1
k1k2E1E2 PF1PE1S1 PF1PE1 PF2PE2S2PF2PE2
PF1PF2PE1+PE2E1E2
E1E2S1S2
(Conics)
(Apollonius of Perga, 262 B. C. ~ 190 B. C.)
(diameters)
111213
K KL HF, FL
K y KLx
H
L
K
GF
y
x
p
HPM
LF pHF (parameter)
(latus rectum) parabola
2y p= x
FX FL FN
OLPX F
HF x FXxMNy
FLpHFd HFFLOLLP
dpxLP LP p x d
= +
p=FL hyperbola
x
p
xd
y
O
XP
L
L LM MO
MO EM EH ON
E ED x EMx LMy EH
pEDdEDEHOXOH dpx
OH OH
2 py px x d
= − ⋅ x 2 py px x d
= − 2 pEH
ellipse
parabola., hyperbola, ellipse
(fall short) (exceed) (fit) elleipsis, “defect”hyperbole, “excess”parabole, “a placing beside” parameter
11 FH 12 FL 13 EH “the straight lines drawn ordinatewise to the diameter are applied in square”
upright side latus rectum
HPM
a : b :: c: d
:: analogia M. Fried
(figure) “upright side”
2
= ±
(the straight line cut off by it on the diameter beginning from the section’s 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.) ABC
ABC [I. 3]DE
ACFFHFG
BC : BA, AC:: FH : FA
K K KL DE KL HF, FL
HPM
(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) ABC
ABC [I. 3]DE
HAKAFGBCKFFG
FL KA : BK, KC : : FH : FL M M MN DE N NOX FL HL X LOXP FN MN FX (applied to) FL FN LX HF, FL 13
(subcontrariwise)
(the parameter)
(And let such a section be call an ellipse)
HPM
(subcontrariwise)DE
FGBCED [I. 7 and Def. 4]E
EHEDAAKED AKBK, KC : : DEEH L L LM FG LM (applied to) EH EM
DE, EH
Apollonius (1952). Conics (tr. R. C. Taliaferro), in Great Books of the Western World, Encyclopaedia Britannica.
The Philosophical Works of Descartes, translated by E. S. Haldane and G. R. T. Ross (1968). London: Cambridge at The University Press.
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”, Science in Context 16(3).
Katz, Victor. J., (1993). A History of Mathematics: An Introduction. New York: HarperCollins College Publishers.
Grattan-Guinness, Ivor, (1997). The Fontana History of the Mathematical Sciences. London:HarperCollins College Publishers.
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).
HPM
1623 6 19 (Clermont-Ferrand)1662 8 19
(Etienne Pascal, 1588-1651)
(Mersenne)
(Torricelli Evangelista,1608-1647)
(Pierre de Fermat, 1601-1665)
(Euclid) (The Elements)
1654 11
HPM
(Christian Huygens,1629 - 1695)
(Chevalier De Mere, 1607-1684)
< )……
n (x+y)
(Traite du triangle arithmetique) 1665
(Omar Khayyam, 1048-1131)
n
1654
(Isaac Newton, 1643-1727)
n ( )nx+1 x
(Pappus)
A” B” C” 2.
1646 1651 1654
1654
1658 (Traite General de la Roulette) 1655
(Pensees)
1. PDF e-mail suhui_yu@yahoo.com.tw
2. 3. e-mail
suhui_yu@yahoo.com.tw 4. http://math.ntnu.edu.tw/horng/letter/hpmletter.htm 5.
HPM ( Boston Consulting Group)
()
()
()
()
1910 11 12 1985 6 12 4
50
1900
1952
189319691896197319001979
Jacques Hadamard, 18651963Norbert Wiener, 18941964
1933
1935 1936
19112004
19101970 1939 1941
20
1946 2 5 9 1926
1924
HPM
1947 1950 3 16
(189219781952 7
19301927
1950193319961927
1983 10
California Institute of Technology
1984 4
1985 6 3 6 12 4
F.Shiratori
HPM
Critical path method, CPMProgram evaluation and review Technique, PERT
1964 18931976
3 18
——
……
—
HPM
2008
8 1999
30
2001
HPM
20
Jean-Claude Martzloff Owen Gingerich
Benno van Dalen
Karine Chemla
Bernd Zimmermann
211
suhui_yu@yahoo.com.tw
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