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M ,
&
.
.
:
.
2007
30-4-2007 , : 1) . ..
( )
2) . .. 3) . ..
mpeira mn poie tn ana mn poreesqai kat tcnhn,
peira d kat tchn.
2006
.
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.
, :
,
,
, .
.
.
,
.
, 2007
:
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.
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,
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.
............................................................................................................... 1
1 ................................ 4
1.1 ........................................................................ 4
1.2 ............................................... 9
2 ................................... 13
2.1 - ......................................... 13
2.2 - ................................................................. 17
2.3 .......................................................... 22
2.4 ................................. 27
3 ............................................ 33
3.1 ........................................................................ 33
3.2 ..................................................... 48
3.2.1 .................................................................... 48
3.2.2 ..................................................................... 54
3.3 ........................................... 72
3.4 .............................. 87
3.5 ........................ 97
3.5.1 ........................................................................ 97
3.5.2 ....................................................................... 99
3.5.3 .................................................................................. 101
3.5.4 ................................ 103
3.5.5 .............................................................. 105
4 ... 109
4.1 ................................................... 110
4.2 ........................................ 112
4.3 ............................... 119
4.4 .............................. 127
5 ...................................................................................... 130
6 ........................................................................................... 133
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(Davis & Hersh, 1981, . 134).
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(, 1993).
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Hughes, (1996, .76),
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3
1
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, (Shinart et al, 2003).
,
. Russell Tractatus
Logico Philosophicus (Wittgenstein, 1978),
.
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, .
1.1
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4
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(, 2001).
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( Gestalt),
5
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,
(, 2000). ,
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,
.
(, 2001).
,
(bottom-up).
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( .. )
.
, ,
, ,
,
.
,
(top - down).
.
- ,
(, 2001).
6
.
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,
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, G.
Frege.
,
. ,
(, 2001).
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7
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(
) (, 2001). ,
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, , (Shinart et al, 2003).
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8
, , .,
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1.2 ,
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,
(, 1985). ,
: , ,
(, 2001).
Chomsky
, ,
(, 2001).
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,
.
Wittgenstein (1978, . 54-65)
:
3.11 ()
.
3.202 .
9
3.323
, .
3.326
3.327
4.002
.
.
:
4.011 - ..
- .
( )
.
, .
,
(, 2001),
,
, ,
(Nirenbourg, 2001). ,
,
(Nirenbourg, 2001).
,
(, 2000).
Frege ,
, .
.
[Zeichen, sign], [Sinn, sense],
[Bedeutung, reference],
10
--. F. de
Saussure (1916), (2001),
, ,
.
,
(, 2000).
,
. F. de Saussure (1979),
(2000),
.
.
,
.
,
(, 1993). Wittgenstein (1978) ,
-
. , +
, (, ), (
, ), , (), ...
, ,
,
. ,
,
,
(Nirenbourg, 2001).
.
Dummett (1977),
(1993),
(). ,
11
Dummett, Wittgenstein
.
,
,
. ,
,
.
,
.
, ,
.
.
, ,
,
, .
, ,
,
,
a priori, (Davis & Hersh, 1981,
. 314). ,
,
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12
2
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.
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18 ,
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19 . ,
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.
13
2.1 -
,
.
.
, .
, ,
(, 2005, . 110).
, , .
,
,
(Heath, 2001,
Cajori, 1919),
, , ,
10
(Boyer, & Merzbach,
1997) , ,
. ,
,
, Nesselmann
Vieta
(Spangolo, 2006, Eves, 1981).
,
, (, 1991)
- .
-
, .
,
14
, . ,
-
.
, ,
. ,
.
(, 1991,
. 38). Frege
(, 1991, . 67). ,
,
.
,
. Girolano Cardano
1545, Ars Magna
, m
,
.
(, 1985, Cajori, 1928).
19 ,
,
, ,
. ,
.
15
(, 985b). Descartes
1701, ,
,
(, 1991, . 35), Wronski 1803
(Davis & Hersh, 1981, . 75), Lagrange
1800 (, 1999), Jacobi 1841,
man muss immer generaliesieren, (Davis &
Hersh, 1981). ,
, Dedekind
, ,
Cantor
(, 2005),
Felix Klein
19 (Davis & Hersh, 1981).
Leibniz,
(caracteristica universalis)
,
. , ,
Leibniz (, 1985).
Bell (1992, . 1, . 195), Leibniz, ,
De arte combinatorya,
.
, ,
, ,
(, )
, .
. Frege Leibniz ,
16
. , calculus
ratiocinator,
(, 1991, . 68).
Leibniz,
,
(Bell, 1992).
Leibniz,
.
Boole 1848
(, 2005) De
Morgan, Jevons, Pierce Schroeder
(Boyer, & Merzbach, 1997). Russell ,
19 ,
Boole
(Boyer & Merzbach, 1997).
17
2.2 -
18 ,
,
.
Euler,
D Alembert Lagrange. ,
,
. ,
19 .
Cauchy, Gauss, Dirichlet, Abel Bolzano
(, 2005).
Peacock
1830,
. Boole ,
, :
, ,
... (Boyer & Merzbach, 1997, .
648). ,
,
, .
19 Frege Russell
(, 2005).
,
.
18
(1991, . 64) ,
, ,
,
Tarski. ,
() , ()
., .
,
, .
,
, ,
(, 1985).
Peano Frege 19 ,
,
.
. ,
,
20 .
Schroeder
,
.
,
, (,
1993, . 75).
, ,
, (1991)
211 [ Jourdain, (1912, .238)]
19
Russell (1908) :
.
. ,
.
Frege :
;
Russell
/ . ()
;
. .
,
.
,
.
.
.
.
,
,
, Frege
ber Sinn und Bedeutung,
=
= ,
.
[Sinn, sense] [Bedeutung,
reference]. .
, Frege
20
,
,
, =.
,
(, 1985).
- Wittgenstein (1978) ,
() .
(
) (.3.32, 3.321).
(.3.325).
,
, ,
, .
20
.
Frege Russell
.
(1993), Frege,
, , Russell,
(. 32)
,
.
. ,
Brouwer, -,
.
,
. ,
(Davis &
Hersh, 1981, , 1993). ,
21
,
. ,
.
(,
1985, , 1993)
, Hilbert
,
, ,
(, 1985).
Gdel 1931 , Hilbert
,
(, 1993). ,
,
Bourbaki. ,
,
, .
,
.
Gdel 1930,
. ,
,
(, 1985).
,
, , ,
, ,
, .
22
2.3
20 , ,
.
,
.
, (,
1991). ,
(, 1993).
,
,
,
. ,
, .
, ,
,
. ,
.
.
,
(Davis & Hersh, 1981, .144).
,
www.math.uoa.gr/me,
Latex:
23
%Here begins the document!
\begin{document}
$\NN,\ZZ,\QQ,\RR,\CC,\TT,\DD$
$\grA,\grB,\grG,\grD,\grE,\grZ,\grH,\grU,\grI,\grK,\grL,\grM,\grN,\grJ,\grO,\grP,\grR,\grS,\grT,\grY,\grF,\grX,\grC,\grV$
$A,B,C,D,E,F,G,H,I,J,K,L,M,N,O,P,Q,R,S,T,U,V,W,X,Y,Z$
$\gra,\grb,\grg,\grd,\gre,\grz,\grh,\gru,\gri,\grk,\grl,\grm,\grn,\grj,\gro,\grp,\grr,\grs,\grt,\gry,\grf,\grx,\grc,\grv$
$a,b,c,d,e,f,g,h,i,j,k,l,m,n,o,p,q,r,s,t,u,v,w,x,y,z$
Two ways to turn from English to Greek
\gt
\tl{Some English}, \tg{ }
\textbackslash \tl{tl} \textbackslash \tl{tg}
\textbackslash \tl{lt} \textbackslash \tl{gt}.\\
(
):
$f: \RR \arw \RR$
$\an$
$\ann$\\
,
:
,
. ,
. ,
24
, (, 1991).
,
,
.
Peano,
.
Boole
Schroeder , Frege (,
1991).
Frege,
(),
,
.
.. (a+b)c=ac+bc.
(, 1991, . 188). , Russell
Frege,
. Russell, (1985, . 232),
,
.
(1985, . 232)
,
.
ermelo-Fraenkel yx ( )yzxzz .
25
Principia Mathematica Russell Whitehead,
.
,
,
, Quine (1967) (1993,
. 81). , ,
(, 1993).
Principia Mathematica Russell Whitehead.
,
. ,
, , (Davis &
Hersh, 1981). , Tomae
26
(game formalism) (1985, . 242),
:
,
.
,
.
,
.
Tomae (, 1985, . 242),
a priori
. ,
,
(1985) .
,
(, 1985, . 243).
, ,
. ,
.
.
,
,
(Davis & Hersh, 1981, , 1985).
,
,
. ,
,
,
(Davis & Hersh, 1981 . 135).
27
2.4
.
. ,
, , ,
, .
,
.
, ,
, .
Hermann Grassman, 1844
Die Lineale Ausdehnungslehrer.
. ,
,
(Davis & Hersh, 1981).
,
. 16 ,
.
Nesselmann
,
(Eves,
1981, Spagnolo, 2006). Vieta Decartes
, ,
, ,
(Eves, 1981, Cajori, 1919).
28
, ,
,
.
, .
() , () ()
(Boyer & Merzbach, 1997)
. , 10841169
,
, MLXXXIV
MCLXIX , ,
. ,
, Van der Waerden (2003, . 44) :
.
.
Newton Leibniz. , 0/0
.
Euler,
,
. Euler
.
Leibniz,
18 (Boyer, 1959, . 242).
Leibniz,
, Newton
,
,
(Bell, 1992).
,
29
,
Leibniz, (Boyer & Merzbach, 1997).
,
,
.
Bombeli 16
x2=1
. ,
(, 1985,
. 241)
, , (pattern)
. Oresme ,
,
,
. 6443 = 864 2/1 = 84 211 =
: (Cajori, 1919, . 127).
,
, ,
2x .
(Boyer & Merzbach, 1997).
Boyer & Merzbach (1997),
Newton, ,
.
Cardano Pascal,
Descartes,
211p
1p 21 4 4
30
. Newton,
, Wallis.
,
x2+y2=1 x2-y2=1
18 , Vincenzo Riccati
. Lambert sinh(x)
cosh(x) tanh(x)
(Boyer & Merzbach, 1997).
,
,
. 18 ,
,
17 ,
(, 1999, . 33).
,
, ,
(, 1999). , 19 ,
(, 1991). Boyer & Merzbach (1997),
Hamilton 1834,
,
,
.
, .
(Davis & Hersh, 1981), Spinoza more
geometrico, , Ethica (Ordina
31
Geometrica Demonstrata), (Bell,
1992).
,
.
,
. Daniel Bernouli
,
.
,
.
dm=K(dp/p), m , p
. (Boyer & Merzbach, 1997)
,
, ,
,
.
.
Fregge (, 1991, . 67)
...
Russell (Wittgenstein, 1978, . 35)
, .
...
32
3
,
.
, ,
,
. ,
,
.
3.1
, ,
, .
, ,
, .
,
. ,
, ,
, ,
,
(Struik, 1966).
, Wedda,
. , ,
, - sticks (Menniger, 1969).
, ,
.
,
33
, .
,
(Struik, 1966).
,
, Damara,
(Menniger, 1969). ,
,
, (, 1999, Menniger, 1969).
, Menniger (1969)
, 4 ,
. ,
,
( 1). ,
.
1. ,
(Menniger, 1969, . 17).
,
,
. ,
,
, , ,
(Struik, 1966).
, . ,
, ,
.
, ,
34
, .
,
(ordering and grouping),
. ,
(Menniger, 1969). ,
,
.
Menniger, (1969), (4, 412-413),
atr pn psaj pempssetai d dhtai, lxetai n mssVsi, nomej j pesi mlwn.
,
. , Dene-Dinjie
, ,
20
, 100 (100=5x20) (Menniger, 1969),
(Hughes, 1996).
18 (Menniger, 1969).
, ,
. ,
, , 5, 10
20. , 18 ,
- achtzehn, - - duo-de-
viginti, (Menniger, 1969).
,
, ,
20 60 .. 12 (-
). , -
, .
, ,
35
, ,
. ,
.
, (abstract concept)
(Menniger, 1969).
,
,
, . ,
.
, , ,
, ,
. ,
(Struik, 1966).
. ,
, .
,
(Hughes, 1996, Struik, 1966).
,
.
,
. 1 5,
, ,
(Hughes, 1996).
, ,
36
, ( 2),
(Struik, 1966, Van der Wearden, 2003).
2. (Menniger, 1969, . 42)
. ,
, ,
,
( 1).
8 VIII (5+3)
9 IX (10-1)
19 XIX (10+9)
1. 8, 9 19
,
, .
,
1/ (Van der Wearden, 2003, Struik,
1966).
37
, .
, 1/12 (Van der Wearden, 2003).
,
,
. , Cajori (1928),
, .
, ( 3),
, 800 .. ,
(Cajori, 1928).
3. , ,
(Cajori, 1928, . 12)
, .
,
, , .
. ,
, .
, ,
38
,
, (Van der
Wearden, 2003).
. -
.
-,
, .
15 ,
, -
. , ,
-
, 400 (Eves, 1981).
,
,
.
.
.
,
, 60
. ,
, (Van
der Wearden, 2003). 1 60
60 ( 4). Van der Wearden (2003,
.33), , .
4. (Van der
Wearden, 2003, . 33)
39
,
. ,
1, , 10. 35
. ,
, 60 1,
, 10 10 x 60.
. , ,
1, 60 , 602 1/60 1/602 .
,
. 60
. ,
, 60+20+4=84 1+24/60=84/60
.
,
(Van der Wearden, 2003,
Bunt, 1981). ,
60, .
, ,
, . ,
, 1x602+0x60+4=3604
1x60+4=64 (Van der
Wearden, 2003).
, ,
. , ,
,
( 5). , ,
, , , ,
, . ,
(Van der Wearden, 2003, Heath, 2001).
40
5. , (Va der
, 24
=1 =10 =100
n
Wearden, 2003, . 42).
,
( 2).
,
999,
9
. ,
.
1305 , 3.000,
4.352. Cajori (1928)
, , .
, ,
=2 =20 =200 =3 =30 =300 =4 =40 =400 =5 =50 =500 =6 =60 =600 =7 =70 =700 =8 =80 =800 =9 =90 =900
2.
,
(Heath, 2001).
41
=10.000,
,
M 250043 (Van der Wearden, 2003).
,
(Heath, 2001).
, ,
. ,
, (), () (). ,
,
, 10,
. , 3 4 7,
, 30 40 70 (Hughes, 1996).
, Van der Wearden (2003, . 43)
,
. ,
, Boyer & Merzbach (1997),
,
. 3520 .
. ,
221 .., ,
(Lam Lay-Yong, 1986, Cajori,1928).
( 3),
,
, ,
, , :
42
1 2 3 4 5 6 7 8 9
3. (Lay-Yong, 1986).
Shang
,
, ,
, .
94571 . ,
,
(Lay-Yong, 1986).
,
. ,
.
.. ,
.
, ,
Kharosti (Boyer & Merzbach, 1997, . 237).
6. Kharosti (Menniger, 1 9, . 50).
Kharosti
Brahm
, 10
96
i ( 7),
, ,
43
,
(Van der Wearden, 2003,
Menniger, 1969).
7. Brahmi (Menniger, 1969, . 395).
Gvalior
,
Brahmi,
(Menniger, 1969).
, Freudenthal, Van der Wearden (2003)
,
. 0
Menniger, (1969) Cajori (1928),
o-
Brahmi
( 7). ,
. ,
,
(Van der Wearden, 2003).
,
,
,
(Van der Wearden, 2003, Boyer &
Merzbach, 1997, Lay-Yong, 1986, Struik, 1966).
Gvalior, () (gubar),
()
. Brahmi,
44
8,
(Menniger, 1969)
8.
, Menniger, (1969, .418)
, Menniger, (1969,
. 4
-
06) (The Indian
numerals in Arab hands). ,
,
(Cajori, 1928,
Struik, 1966, Lay Yong, 1986). ,
Severus Sebokht ( )
622 ... ,
. ,
. ,
, 595
45
346 (Boyer & Merzbach, 1997, Struik, 1966)
,
(),
(Menniger, 1969).
,
. , 8
9 ,
. ,
, (Menniger, 1969, Van der
Wearden, 2003). -
,
Gerbert ( ) 10
. , 12 , ,
Al Khwarizmi. ,
, Algorithmi de numero Indorum,
Hisab aljabr walmuqabala. ,
(Boyer & Merzbach, 1997, Van der
Wearden, 2003, Menniger, 1969, Struik,1966).
, Leonardo di Pisa Liber abaci,
1202
, .
, , 0,
- (Boyer & Merzbach, 1997).
, o ,
, ,
Leonardo di Pisa. , ,
- (Boyer & Merzbach, 1997, Vogel, 1996).
46
-
, , ,
(Van der Wearden, 2003).
IImIIIcXV
2315 (Struik, 1966). ,
- .
400 ,
, . ,
, ,
, Adam Riese ( 9) .
9. Adam Riese (Menniger, 1969 .
437 & 35 )
15
,
2
-
. - ,
.
Van der Wearden, (2003, . 55)
,
Menniger, (1969, . 55),
, , .
47
3.2
, .
,
, .
3.2.1
.
,
, .
,
, .
,
3, 4 7 (Struik,
1966).
,
. , ,
, , ,
(, 2003).
,
(, 1975, Van der Waerden, 2003).
,
- , .
,
17 (, 2003).
,
,
48
.
, ,
, . ( 10)
1342, x2+10x=39
, .
10.
(Hairer et al., 2000)
16 .. ,
, , . x2, x3, 6x
3 .
, Cardano Ars Magna
, x3 + 6x =20,
20 (Boyer & Merzbach,
1997).
,
(, 2005),
.
,
, , , , .
. ,
- .
,
.
.
49
, 1
Estw doqesa eqea peperasmnh AB. De d p tj AB eqeaj trgwnon spleuron sust- sasqai. KntrJ mn t A diastmati d t AB kkloj gegr- fqw BGD, ka plin kntrJ mn t B diastmati d t BA kkloj gegrfqw AGE, ka p to G sh- meou, kaq' tmnousin ll- louj o kkloi, p t A, B shmea pezecqwsan eqeai a GA, GB. Ka pe t A shmeon kn- tron st to GDB kklou, sh stn AG t AB plin, pe t B shmeon kntron st to GAE kklou, sh stn BG t BA. decqh d ka GA t AB sh katra ra tn GA, GB t AB stn sh. t d t at sa ka llloij stn sa ka GA ra t GB stn sh a trej ra a GA, AB, BG sai lllaij esn.
, ,
,
,
. 1342 ( 11),
,
2 .
11. (Hairer et al.,
2000).
Leonardo di Pisa Regiomontanus
c f, Stevin 1634
, , , , . Leibniz 1676
,
1B, 2B, 3B, 1D, 2D, 3D 1b, 2b, 3b
50
. Leibniz,
. 12,
Leibniz .
12. Leibniz (Hairer et al., 2000).
Euler, ,
x, x, x y, y, y,
,
a, b, c, A, B, C,
, . , r, R
s ,
, . 4rRs=abc
,
Euler, (Boyer
& Merzbach, 1997).
,
, , , .,
,
. , ,
.
(), .
51
150 .. , ,
, .
, 5 .. , , ,
(Cajori, 1928).
==
, , , , , . .
Maurolykus 1555 , , ,
. 1634 Herigone Cursus mathematicus
:
< , , , , _| , = , , , , , , 5< 6
(identity).
, -
is simil. (Cajori, 1928).
( 4),
Cantor 1801 De la Correlationdes figures de
geometrie Cajori, (1928) :
A, B, C,
, ,
BCD
CDAB
F C D AB
F
AB CD
= | =
AB BC
AB BC
ABC ABC
ABC BC A
ABC ABC
4. Cantor 1801 (Cajori, 1928, . 422).
ABCD
ABC
CD
BCD
53
3.2.2
, ,
18 .
,
, .
,
. 18 ,
,
. ,
,
.
, ,
, ,
.
,
.
: Ka t farmzonta p' llhla sa llloij
stn. Frege, ,
Leibniz: Leibniz ,
.
.
Leibniz . [dasselbe]
, [gleich]
(, 1993, . 52).
, Russell :
x=y.=():!x. !yDf. , x y
x,
y (, 1993).
15 ,
54
, . ,
(it
gives) (Cajori, 1928).
,
aequales, aequantur, eguale, esgale, faciunt, ghelijck, gleich
(Cajori, 1928). ,
. , (
) ,
.
pha phala,
Al-Qalasadi
(Cajori, 1928). Regiomontanus
Pacioli ( 13), ()
, Cardano
(Cajori, 1928).
13. , ,
Summa Pacioli (Cajori,1928, . 110).
= Robert Record 1557
The Whetstone of Witte, ( 14),
, (Cajori, 1928).
:
55
(Eves, 1981, , .
150).
, ,
,
(Cajori, 1928).
,
, [ , || , | ,
2|2 Herigone (1634), . Herigone
3|2 2|3 .
|| Wilchelm
Holzmann, Xylander 1571
. , Cantor
, Xylander
(Cajori, 1928).
14. Robert Record
(: http://members.aol.com/jeff94100/witte.jpg)
= ,
,
56
ae aequales,
(Cajori, 1928). ,
Descartes (1637) ,
, Huygens 1646, . Rolle
1690 Jac. Bernoulli 1713 (Cajori, 1928).
=, .
Boyer & Merzbach (1997),
,
.
1557, 1618
Artis analyticae praxis
T. Harriot, Clavis mathematicae W. Oughtred Trigonometria R.
Norwood (Cajori, 1928). ,
, , Newton
Leibniz ,
(Cajori, 1923a, Cajori,
1928). 18 ,
. Jac. Bernoulli,
Ars Conjectandi, 1713 ( 15),
(Cajori, 1928).
15. Jac. Bernoulli (Hairer et al., 2000).
57
,
=
. Record, ,
(Eves, 1981).
> <
T. Harriot Artis analyticae praxis 1631, (Cajori, 1928,
Boyer & Merzbach, 1997), Oughtred,
, . O Isaac Barrow (1630-1677),
Lectiones Opticae & Geometricae (London 1674),
: A major est quam B A
minor est quam B (Cajori, 1928).
,
. ,
.
, ,
( 16),
.
16. (Cajori, 1928)
, ,
,
. leyij parxin,
, .
, ,
.
.
74-1/14 (Heath, 2001).
, Arithmetic Bakhshali 8 10
.. , + .
12 7+ 12-7
yu, yuta,
(Cajori, 1928). , ,
58
. 3 4
7 3 4 . , 15 ,
, , ,
(Cajori, 1928).
1
, + -,
, Mercantile Arithmetic
Behende und hpsche Rechenung auf allen Kauffmanschafft Johannes Widmann,
1489. ,
,
. Widmann
( 17), "4 centner + 5 pfund" "5 centner - 17 pfund,"
"Was - ist / das ist minus ... und das + das ist mer."
(Cajori, 1928).
17. Widmann, ,
Augsburg 1526. (: http://members.aol.com/jeff94100/widman.jpg)
59
+,
et.
15 , 1486,
et . Cajori,
et ,
+ ,
t ( ) ,
1417. ,
Widman ,
, (Cajori, 1928).
Widman,
+
.
Henricus Grammateus Rechnenbuch 1518 Rudolf
Coss 1525, + ,
(Cajori, 1928). , +
Robert Recorde The Whetstone of Witte, 1557 (Boyer
& Merzbach, 1997). , Paciolli
Chuquet, 1484, ,
piu minus, p p~
m (Cajori, 1928, Boyer & Merzbach, 1997).
Regiomontanus 1460,
m~
9i ( 18). ,
(florescentform) (Cajori, 1928). m~
18. Regiomontanus (Cajori, 1928).
60
16
+ ,
17 (Cajori, 1928).
p m,
+ Rudolf (1525)
Stiefel (1544) (Boyer & Merzbach, 1997), Vieta,
,
| ( 19)
(Cajori, 1919, Cajori, 1928).
19. Vieta (Hairer et al., 2000, . 7)
+ ,
,
, .
? , . . ,
, . , ,
, .
, Herigone ~, Descartes Geometrie
. 18 ,
1---R---11 11--1 (Cajori, 1928).
61
+ , ,
,
.
,
. ,
, +a a a
. Wolfgang Bolyai 1832
+ (Cajori, 1928).
Albert Girard 1626,
ou, ou+
, .
, 1631 Oughtred Cavis mathematicae,
, Wallis,
Jones . , Schooten ,
Newton , , . , Euler, 1755 (Cajori, 1928).
m
,
.
, (-DU),
. , Rhind,
. ,
,
. 8, 9 10 ,
Bakhshali,
(indicated by placing numbers side-by-side). al-
Qalasadi 15 ,
(Cajori, 1928).
Bahaskara ,
bha,
, . Vieta,
, A in B Stifel, 1545,
62
Deutsche Arithmetica (M)
D (D) .
Stifel Stevin, 3xyz2 31 sec1M ter2 , sec ter , . 1553, Stifel Coss
Rudolff,
.
,
Herigone,
, * Rahn Banker
Teutsche Algebra Rahn,
17 (Cajori, 1928).
, ( Cajori (1928)), William Oughtred Clavis
Mathematicae - - 1631
1647. Oughtred .
John Wallis Seth Ward.
Oughtred ,
150 ,
. ,
. ("Appendix to the Logarithmes")
Descriptio John Napier, 1618,
.
Oughtred. ,
, , ,
(Cajori, 1928, . 251):
315172 295448
395093 174715
63
,
,
, (Cajori, 1928).
(.) Leibniz,
Oughtred,
x . John Bernoulli
: "I do not like (the cross) as a symbol for multiplication, as it is easily
confounded with x; .... often I simply relate two quantities by an interposed dot and
indicate multiplication by ZCLM.". Leibniz, Harriot Artis analyticae praxis 1631
aaa-3bba=+2ccc (Cajori, 1928). Cajori (1928) , Harriot,
Gibson 1655 Syntaxis mathematica,
,
. Leibniz
De arte combinatoria 1666, C
.
18 , Cristian Wolf,
Euler (Cajori, 1928).
, ,
.
*, , .
Rene Descartes, Gullberg (Miller, 2004). Cajori (1928),
Gauss, 1812.
, ,
,
. ,
64
,
, . ,
% ,
o
M , % o
M , (60x2+2520)/(x4+900-60x2) (Heath,
2001). ,
, .
,
(Boyer & Merzbach, 1997), Bakhshali
bh bhga
(part)(Cajori, 1928). al Hassar 12 ,
Leonardo di Pisa
, Liber abbaci, (Cajori, 1928).
16 /
, 1845, De Morgan (Boyer &
Merzbach, 1997).
,
Arithmetica integra Stifel 1544 Oughtred
Clavis mathematicae : 24:8 8)24 8)24(.
Stifel Stevin, ,
D (D) . Rahn Teusche Algebra,
, . Rahn ,
, Wallis
Barrow, ,
, (Cajori, 1928).
Leibniz, , De arte combinatoria
C . 1684, Acta eruditorium, : , x:y x
y yx . ,
65
(Cajori, 1928).
,
, : 36)116(3,
116:36=3, : 116)36(3 Bezout,
1833, :
14464 8
1808
Miller (2004),
. ,
1882, Complete Graded Arithmetic James B. Thomson,
,
( 20).
20.
, 1888
The Elements of Algebra G. A. Wentworth
( 21).
.
21.
, .
,
66
, . ,
. 16o ,
,
.. Radix, quadrata
R ra, radix , latus (Cajori, 1928).
,
ka, karana
(Eves, 1981).
l
, 1220
Leonardo di Pisa Practica geometriae,
(Cajori, 1928).
, Ars Magna Cardano. Chuquet
2 ,
(Boyer & Merzbach, 1997, Cajori, 1928),
Bombeli Rq Rc
(Eves, 1981). ,
Q cu C . 1525 Die Coss Christoff Rudolff
,
(Cajori, 1928). ,
r,
radix. Institutiones calculi
differentialis (1775) Euler, Cajori (1928), ,
, ,
. 1637, Descartes La Geometrie,
, ,
.
Albert Girard 1629 Invention nouvelle,
3).20+392 3 39220+ (Cajori, 1928). Stevin,
67
(Boyer &
Merzbach, 1997) e (Cajori, 1928). Cajori (1928), Girard
John Wallis,
320
3 20 Michel Rolle (1652-1719), 1690
Trait d' Algbre .
. ,
, Newton
(Cajori, 1919).
,
.
,
.
, , .,
.
, , ,
.
u v,
Cajori (1928) Eves (1981), Pacioli Summa,
1494 1523 : R7pR14, 7 14+ , R (radix universalis niversalis)
.
,
.
Cajori (1928), Le Triparty en la Science des Nombres
Chuquet 1484. Chuquet
14 180+ . , Alexander, Bombeli (Cajori, 1928).
68
Frans van Schooten,
Vieta , . ,
1646 ,
B(D2 + BD) Vieta
in . + Din Bquadratum D.
.
Newton, Analysi per Aequationes mumero terminorum Infinitas,
4 5 12 17 0y y y y + + = {[(y-4)y-5]y-12}y+17=0. , 1631 Hariot,
(Cajori, 1928).
, 16 17
Tartaglia Cardan, Ars magna,
Opera 1663, Stifel
. [ ]
Algebra Bombelli, 1550 (Cajori, 1928). Bombelli,
7 14+ R[7pR14] (Eves, 1981). Cajori (1928), 1593 Zetetica
Vieta. 18 , ,
,
,
.
, ,
, ,
. 1202 Liber Abaci Leonardo di Pisa,
.
.
,
Cajori (1928), 1795, Charles Hutton
"Numeration" Mathematical and Philosophical Dictionary.
69
,
16 (Cajori, 1928).
Vieta .
141.421.356.24 ,
314.159.265.35
1.000.00 314.159.265.35
(Boyer & Merzbach, 1997). 1585 Stevin
La disme ( - The Tenth) (Arcavi et al 1991, Struik, 1966).
16 , ,
,
, 1616, Descriptio Napier
(Boyer & Merzbach, 1997).
,
.
, ,
e, , i . Wallis , Barrow
Jones
. ,
Euler (Eves, 1981). Euler, Boyer & Merzbach
(1997),
. e,
,
i ,
. i,
Gauss, . , e i
Euler ,
0 1 ei+1=0.
Euler, f(x) x,
1734 1735 (Boyer & Merzbach,
1997).
70
13 ,
. ,
.
,
, -
, , ,
. ,
Vieta .
,
.
71
3.3
,
:
.
,
.
(Eves, 1981).
,
, ,
.
,
1700 ..,
(Van der Waerden, 2003).
,
, .
,
, .
, ,
,
xy x-y, (Van
der Waerden, 2003). Van der Waerden (2003),
,
, , ,
. ,
,
,
.
, ,
. (,
, 2):
72
'Ariqmj d t k mondwn sugkemenon plqoj, (, 1975)
,
, ,
. ,
, ,
.
, ,
, ., 1+3+5++(2n-1)=n2
, x2=(-x)
(Van der Waerden, 2003).
, (,
, .)
,
.
, Nesselmann
(Spangolo 2006, Eves, 1981, Heath, 2001).
, ,
17 .
, ,
,
(Boyer & Merzbach, 1997).
,
,
, . Bashmakova
(1999a),
Vieta.
3 .. ,
. ,
73
, ,
, (Eves,1981).
, 189 .
,
,
.
,
(, 2003, Eves, 1981).
, al-
Khowarizmi (Van der Waerden, 2003). Heath (2001),
,
. , Eves (1981)
, , (2003)
Bashmakova (1999a),
,
, , .
. , .
= ( )( )( ) , .
,
(Bashmakova 1999a). Heath (2001),
,
.
,
,
(,
74
2003). , ,
,
, .
,
(Heath, 2001, Van der
Waerden, 2003). ,
(Bashmakova 1999a, Van der
Waerden, 2003). ,
, : , x2, , x3,
x4, x5 .
,
, , 1/x=x-1,
, x-2=1/x2 .
o
M ,
Bashmakova (1999a), .
. ,
2x5 2/x5 .
,
. , ,
,
.
x3-2x2+10x-1=5 :
o
M o
M
,
.
, (2003, . 110),
,
75
. ,
() ,
() .
, al-Khowarizmi,
al-jabr al-muqabala, al-jabr
algebra - (, 2003).
,
. Bashmakova (1999a),
()=.
, VI14, 90-15x2
54 15x2-36.
-15x2 . ,
(intermediate
computations), Bombeli.
, . Nesselmann
(Klein, 1968, Eves, 1981).
, Hultsch,
(2003, . 108-109),
, .
,
. ,
,
(determinate and indeterminate) .
Descartes,
, (Bashmakova, 1999a).
Vieta, Descartes, Fermat
17 , J.
Klein:
Vieta .
76
Vieta (,
2003, . 110). , Bashmakova (1999a),
Poincar,
20
(algebraic curves), .
,
. ,
. ,
. , Bhaskara x2-
45x=250 50 -5 (Struik, 1966). , Brahmagupta 7
ya yavattavat
ru rupa
. ,
, ka
kalaca , ni nilaca , .
yu pha (Eves, 1981,
Cajori, 1928). , pha 12 yu 1175
1217
15 =+
(Cajori, 1919). Bakhshali
sunya . sunya
(empty) ,
. ,
,
(Cajori, 1919).
.
al Khowarizmi,
825. ,
, ,
.
, , ,
(Struik, 1966, Eves, 1981, Cajori, 1928). ,
77
,
,
Horner.
, (Struik,
1966).
,
(Heath, 2001).
,
,
, .
,
,
, . ,
888 ..
. ,
,
,
=4, =5 =6 (Vogel, 1996).
, ,
,
. ,
, Leonardo di Pisa. 1202,
Leonardo Liber Abaci,
, al Khowarizmi,
. T Liber Abaci, ,
-
(Struik, 1966). ,
, ,
,
.
78
, . radices radix
x, x2 census. ,
radices 1221 x
2112 ,
factus ex...
,
. ,
Leonardo di Pisa (Eves, 1981,
Cajori, 1928, , 1996).
, 14 15 ,
.
,
. , Pacioli Summa de Arihmetica 1494
co x cosa
, ce x2 censo, cu x3 cuba, cece
x4 censocenso Stifel,
,
coss, zensus, cubus zenzizensus (Cajori, 1919, Boyer & Merzbach, 1997).
Regiomontanus ,
, .
,
(Boyer & Merzbach,
1997, , 1996). 1484,
Triparty Chuquet. ,
p~ , m~ 2
, 3 . res, chose,
cosa, coss premier .
, , Chuquet
.
5x, 6x2 10x3 .5.1, .6.2, .10.3. ,
79
9x0 .9.0 9x-2 .9.2.m .9.
seconds moins ( ). , Chuquet
.72.1 .8.3
.9.2.m, 72x / 8x3 = 9x-2.
Chuquet .4.1 .2.0, 4x=-2,
(Boyer &
Merzbach, 1997, Cajori, 1928).
m~
16
,
, ,
.
,
. , ,
: Coss Rudolf, Rechnung Apian
Arithmetica integra Stifel. ,
, ,
. ,
+ ,
. Stifel
, Harriot
Artis analyticae praxis 1631 (Boyer & Merzbach, 1997, Cajori, 1928).
, Ars magna Cardano Algebra
Bombeli ,
. ,
Ars magna ,
,
, Ferrari. Cardano
, Summa Pacioli
,
. p m,
. 1545,
80
10 40, 15-5+
15--5 , :
5p: m:15
5m: m:15
25m:m: 15 q~ d. est 40
Ars magna 1663,
:
5. p~ . . .15. m~
5. m~ . . m~ .15.
25.m.m. 15.quad. est. 40.
Cardano
.
,
(Boyer & Merzbach, 1997, Cajori, 1928).
Algebra 1560 Bombeli,
,
. ,
Bombeli
. Algebra
1579,
(Arcavi et al., 1991). Bombeli 1p.5Rm.4, 1 zenus
5 res 4 x2+5x-4, ,
,
(Boyer & Merzbach, 1997, Klein, 1968).
,
.
2x, 4x2 5x3 2 , 4 5 (Boyer & Merzbach, 1997,
Cajori, 1928). , Bombeli
,
3 1 2
81
.
,
Bombeli (Boyer & Merzbach, 1997).
,
Vieta 1591. Vieta
ars analytica
.
, nullum
non problema solvere. Vieta
- In artem analyticam isagoge, 1591
(Bashmakova 1999b). ,
, (Boyer &
Merzbach, 1997, Klein, 1968).
Bashmakova (1999b),
,
,
Vieta. , Regiomontanus Stifel , Leonardo
Cardano ,
Vieta . ,
, . Stifel ,
,
. Vieta,
,
.
,
. Vieta
,
2+CA+D=0, B, C D (Boyer
& Merzbach, 1997, Cajori, 1919). , Vieta
. ,
(homogeneous magnitudes) ,
82
. ,
, longitudo (),
latitudove (), planum ( ), solidum () .( 22).
: quadratum Ein C
3 solido B-cubusA , 3
, C ,
C . 23
cy3b-x x, y
b, c (Klein, 1968).
, Vieta
(cossist) + , =
= |-|. ,
in, applicare,
aequale. , quadratum, cubus
radix l
l. latus
l. 121 121 . ,
Van Schooten. ,
(Cajori,1928 , Bashmakova, 1999b).
22. Vieta 2+2=2 (Hairer et al., 2000).
In artem analyticam isagoge (Cajori, 1928) :
B in A quadratum plus D plano in A aequari Z solido BA2+D2A=Z3
Ad Logisticen speciosam notae priores,
1646
(Bashmakova, 1999b). (Cajori, 1928),
3+32+32+3 :
cubus, + quadrato in ter, + in quadratum ter, + cubo,
83
Vieta,
,
, . Vieta,
,
, .
,
, , ,
- ars analytice. 5
Isagoge, (Law of Zetetic-De Legibus
Zeteticis),
. , logistica
numerosa ,
logistica speciosa. , ,
, .
Vieta, x a, ,
,
(, 1996). Klein (1968)
. ,
- The modern concept of number is born (Klein,
1968, . 175).
, Vieta, ,
.
,
.
, .
,
,
(Klein, 1968, ,
1996). , Vieta
,
,
(Cajori, 1919).
84
. 16 , Stevin
,
Brgi , . , x4+3x2-7x
, Stevin
4 2 1 1 + 3 - 7
Brgi
iv ii i 1 + 3 - 7
Stevin 3/2
(Boyer & Merzbach, 1997, Klein, 1968).
17 ,
( Descartes),
,
Stifel Harriot. T 1634, Hrigone Cursus mathematicus
a, a2, a3, ., 1636 Hume L'Algbre de Vite d'vne
methode novelle, claire, et Facile
iii 3 (Cajori, 1928).
Hume, Boyer (1950)
3 Aiij iii. , Descartes
,
x3 xxx ,
, . Leibniz,
xm ,
Newton
(Cajori, 1928, Boyer, 1950).
Vieta, Descartes
. Geometry
Vieta,
85
. ( -
segments) (). ,
,
,
(Bashmakova, 1999b). Descartes,
(Klein,
1968). , Descartes
,
.
Newton Universal Arithmetic (Bashmakova, 1999b).
Vieta
Descartes Fermat,
.
. ,
, .
,
, -
(potential function)
, (were eventually recognized as forming a new analysis-
the calculus) (Kleiner, 1989, Boyer, 1959, . 98).
86
3.4
Boyer (1959),
,
,
. ,
, ,
,
( .., 1987).
, ,
,
,
, ,
(, 2005).
,
Bourbaki,
Riemann,
(, .., 1987).
800 ..,
,
,
, ,
, . 15
,
(, 2005).
, 16 ,
. ,
Vieta, Decartes
Fermat, ,
, .
87
, ,
(, 2005).
, 17 ,
,
, , -, ,
. ,
Stevin, , Kepler, Cavalieri, Torricelli, Fermat, Pascal Roberval,
Wallis, Gregory Barrow, ,
,
(, 2005). , ,
.
, ,
. ,
, .
, ,
y=xn.
, ,
. ,
, ,
.
Newton Leibniz 1675 (Maor, 2005).
Newton Leibniz,
,
. ,
.
, ,
.., (1987, . 210): ,
,
, ,
88
, ,
. ,
, Newton
Leibniz, . Leibniz,
:
(, 2005, . 59).
Newton, x, y,
(fluent), .
(fluxion) .
Methodus fluxionum, Newton,
,
. x t,
x ,
.
20 1665. , x ,
x(t)
x&
x&
dtdx . x y,
,
. ,
x& y& x&& y&&x& y&
xb
yy
-
xxaa - . ,
, x y,
. ,
Algebra
Wallis 1693, Newton, ,
Quadratura curvarum 1704 (Cajori, 1928, Boyer, 1959).
,
(moment) .
|x
|y
||x
||y
|x
|y
x&
89
y=xn =nxy& n-1 x& . , Newton,
(moment), (conatus) (indivisible), , e, a, o
(Boyer, 1959).
Newton, Principia, x, y, z,
p, q, r , X, Y, Z,
(Cajori, 1928). , Newton
, , , , , x, y, , ,
, , .
Newton ,
,
x&& y&& x& y&|x
|y
||x
||y
xaa
64 xdxaa64. (Cajori, 1928).
, .. (1987), Newton ,
. , ,
- quadratura
, ,
, .
Newton, ,
, .
, ,
x, , , x, x, x.
, Newton ,
, . ,
B. Taylor Methodus incrementorum 1715,
: = r adeoque p = r .
,
, ,
(Cajori, 1928, Boyer, 1959).
|x
||x
p& s& s&
Newton, Leibniz,
,
90
,
,
. (2005, . 68), C.
H. Edwards , Leibniz
, ,
Newton.
Leibniz,
26 1675,
y l dy
. 11 1675,
dx dy x y (Cajori, 1923).
dx dy,
Leibniz, Acta eruditotum 1684 d ad
dx, dx ad dy dx:dy, dydx . ,
, ddx:d 2y
dx3 (dx)3 3dx (Cajori, 1928).
Leibniz d : d, b ax , aeq. ba dx
b bax - (Cajori, 1923). Nova Methodus, Leibniz,
. x y
x, dx x dy
y (, 2005). , ,
calculus differentialis (Struik, 1966).
, Leibniz, Cavalieri
omn.pa=omn.yl=2
2y .
Mengoli Fabri
1659, omn. omnes linea,
.a, (Cajori, 1919, Rickey, 1992).
, .. (1987), omn.,
91
omnes (priores) magnitudines ad
omnes.
, 29 1675,
Leibniz : omn. omn.l, l. , Leibniz
, , :
Satis haec nova nobila, cum novum genus calculi inducant -
, , ,
d (Cajori, 1928, Rickey, 1992).
l
23. Leibniz
(Cajori, 1928, . B, . 243).
Leibniz ,
calculus summatorius- ,
. ,
92
Jacob Bernoulli, Acta eruditorum 1690.
Jacob, Johann Bernoulli, calculus integralis-
. Leibniz, , ,
,
(Cajori, 1928, Rickey, 1992).
Leibniz, 1695, Johann Bernoulli,
= n=-m .
, ,
Nieuwentijts ddx, d
md n
3x,
dddx d4x (Cajori, 1923).
18 , Leibniz
,
. Leibniz
, Euler,
,
(prominence) (Boyer, 1959, Rickey,
1992).
18 , Boyer (1959, . 243),
, Euler
Leibniz . , Leibniz,
Newton, ,
. ,
, ,
, . D Alembert ,
Euler, , ,
, Newton, prime
and ultimate ratios .
,
. D Alembert,
93
,
Leibniz Euler 18 .
,
,
,
(Boyer & Merzbach, 1997, Struik, 1966).
, Lagrange 1797, Theorie des functions
analytiques,
, .
Taylor,
fx, Lagrange
fx, fx, fx y, y, y, y (Cajori, 1923, Struik, 1966).
Arbogast, D
, .
a+bx+2.1
c x2+ a=Fa, b=DFa, c=DDFa .
D c
mDm
Dm
....3.2.1,
dx
xd , 22
.2.1
dxxd , x (Cajori, 1928).
, 1770,
Monge, Condorcet Legendre, Legendre,
1786
x (Cajori, 1928).
, ,
16 ,
. lim.
Simone Lhuilier 1786, : lim.q:Q Lim.xy
.
L Carnot 1797
Cauchy 1821. , Cauchy
. , x
94
, lim.(sin.x) 0, lim.((x1 ))
, lim.((sin.x1 )) . ,
lim. lim. 19 ,
,
. Limx=a,
Weierstrass
Hamilton { } (Cajori, 1923).
1lim
= == nn pLim .
=nlim
,
. G. H. Hardy 1908 )n1lim(
n
, :
Leathem Bromwich, : , , ,
, .
. , lim ,
nlim xlim a xlim =nlim
=xlim a =xlim
u' xu (Cajori, 1923, . 43).
19 ,
, ,
,
. ,
, .
, ,
, ,
17 18 Leibniz,
(Struik, 1966, . 182),
(2005, . 68) ,
,
95
.
,
, (2005, . 68) ,
Leibniz,
, ,
,
, Cajori (1928,
. B, . 181),
, Faust Goethe War es ein Gott, der diese
Zeichen schrieb?
96
3.5
,
. ,
Cantor
, . ,
, +1=, += ., ,
(Davis & Hersh, 1981).
, ,
. ,
, ,
Oresme (Boyer & Merzbach,
1997). ,
.
3.5.1
, 17
,
.
(1999), , ,
,
. ,
,
, ()()=(+) ()(+)=() (Bashmakova 1999a, , 1999). ,
97
Arithmetic Bakhshali 8 10 .. ,
+ . Cajori (1928),
, 12 7+, 12-7.
,
3 (Cajori, 1928). 4
1
, 12 ,
() ,
. Bombelli,
0-1069, -1069,
-1069 (Cajori, 1928).
Chuquet,
, .4.1 m~ .2.0, 4x = 2 (Boyer &
Merzbach, 1997). ,
, . ,
Stifel ,
numeri absurdi, (Boyer & Merzbach,
1997). , Cardano
numeri ficti, .
,
(Boyer & Merzbach, 1997).
, (Vieta, Hariott .),
, Vieta
. ,
1629, Girard, Invention nouvelle en l algbre,
(Boyer & Merzbach, 1997).
1657, Hudde ,
. ,
(Cajori, 1928).
98
,
. , +a
a a . ,
,
n p, . , as ar , , .
19 , + , 10
+10, .
-3,-2, -1, 0, +1, +2, +3, Continuum
Huntington 1917 (Cajori, 1928).
3.5.2
,
.
,
, 13 . ,
Chuquet 4+x2=3x,
. Chuquet : Tel
nombre est ineperible, (Boyer &
Merzbach, 1997, Cajori, 1928). ,
, , Cardano
Bombelli. , o Cardano
15-5+ 15--5 , 5p: m:15 5m: m:15.
Bombelli , , 4.p.R[0 4.~m ]
4-4+
. , ,
. di meno dm.
99
i= 1- . (+2i)(+i)=-2 : p.dm.2.via.p.dm.1 equal. m~ 2 (Boyer & Merzbach, 1997, Cajori, 1928).
,
. ,
, Cardano Tartaglia,
. ,
,
(Boyer & Merzbach, 1997).
a- ,
, , Euler
a- , 1-
. Euler i
, 1777. , Euler
i . Euler
Jean
Bernoulli 1740 (Boyer & Merzbach, 1997, Cajori, 1928).
, , Hamilton, 1837,
(1, 2)=1+2 1- , 1903, Burkhardt (, )=+i (Cajori,
1928). i ,
Gauss Disquisitiones Arithmeticae
1801. , Gauss
,
.
, 1, -1 i, ,
,
directe, inverse, laterale Einheit (Boyer & Merzbach, 1997, Bell, 1992).
100
3.5.3
: () ,
() ,
. ,
,
. ,
. ,
,
-. (notion)
, 18 ,
.
,
.
: () ,
(Bombelli, Stifel
), () (Vieta, Descartes ), ()
, (Kepler, Galileo
) () (Descartes, Fermat
) (Kleiner, 1989).
, , ,
. ,
,
.
, ,
,
- (potential function). ,
, . ,
101
. Newton
Leibniz ,
(Kleiner, 1989, Boyer, 1959).
1692 Leibniz function -
.
1718 Johann Bernoulli.
, Leibniz :
x, x,
x, y, x (Cajori,
1928).
, Leibniz Johann Bernouli
, , n,
B, F, G, C , . x z b,
d, ., : x x z. Euler,
f(x) 1734 Commentarii Academiae Scientiarum Petropolitanae
(Cajori, 1928). , 1748, Euler Introductio in Analysin
Infinitorum,
(Kleiner, 1989).
x 1
x;y x 1 2
r.1 x
3.5.4
,
, .
,
.
, .
102
Regio
, ,
,
, .
,
(Boyer & Merzbach, 1997).
.
montanus, .
, ,
Regiomontanus, Rheticus,
. Vieta,
,
. ,
2-2
)-(2
)(
A
BA
a
ba +=
+
,
1583 Thomas Geometria rotundi,
tangent secant
sin., tan., sec., sin.com, tan.com, sec.com.
(Cajori, 1928).
,
2Bb
Fink
. (Boyer & Merzbach, 1997). Thomas Fink
.
,
17 18 ,
. ,
s, S,
sin, sin., sen., sn , , cs, S.2., Cos., cos. cos,
t, T, tang., tang, tng, tg tan., (Cajori, 1928).
, ,
Introductio Euler.
103
, sin., sin, sn
, cos., cos, cs .,
(Boyer & Merzbach, 1997, Cajori, 1928):
sinx= 1-2
- -1--1 xx ee , cosx= 2
-1--1 xx ee + eix = cosx + 1- sinx .
, y2=1
x2-y2= 1, . 1757,
x2+
Riccati,
Sh.x Ch.x
. , Sc. Cc.
, ,
, Lambert
sinh(x) cosh(x) . ,
,
sinx, cosx eix
:
sinh(x)= 2- -xx ee , cosh(x)=
2
-xx ee + ex=sinh(x) + cosh(x).
,
3.5.5
,
, .
(Boyer & Merzbach, 1997, Cajori, 1928).
.
,
,
. ,
,
104
(, 1989).
,
1
20+21x+22y=0
30+31x+32y=0
,
10.32+11.32x-12.30-12.31x=0
x ,
x
:
a10.a21.a32+a11.a
.
Leibniz De L Hospital, 693. ,
Leibniz
, . ,
3 2 ,
:
10+11x+12y=0
. , Leibniz,
, - .
, Leibniz ,
:
10.22+11.22x-12.20-12.21x=0
:
10.21.32+11.22.30+12.20.31-10.22.31-11.20.32-12.21.30=0
22.a30+a12.a20.a31-a10.a22.a31-a11.a20.a32-a12.a21.a30=0
105
Leibniz :
10.12.32 10.22.31
11.22.30 = 11.20.32
12.20.31 12.21.30
.
, C. ,
2x2, 3x3 4x4,
Mac Laurin, 1729.
Leibniz, (, 1989, Cajori 1928).
,
Cramer,
Introduction a l Analyse des Lignes Courbes algebriques-
, 1750. ,
5
A+By+Cx+Dy2+Exy+x2=0, Cramer 5
5 .
,
. , 2x2 :
y= 12211221
--
ZZAZAZ z= 1221
1221
--
ZZAA
, Cramer
.
Bzout Euler
:
(, 1989).
1771, Vandermonde, ,
. ,
106
111 + 2
12 + 3
1 = 0
2 2 211 + 2 + = 0
2 3
a
b
a
b
.
, ,
2x2 :
=
a
b
- b
a
3x3
Vandermonde, ,
,
( , 1989,
Cajori, 1928).
, . Laplace
ax2+2bxy+cy2, (a, b, c)
b
a b c
a b c ab c ac
a b
, Laplace Lagrange,
,
,
, : 1a, 1b, 1c, 2a, 2b, 2c, .
, ,
. , 3x3
(1a 2b 3c), Binet (, 1989).
determinantem - ,
Gauss, 1801 . Gauss,
Disquisitiones Arithmeticae,
107
Cauchy Binet,
1815, determinant
bb-ac. , Lebesgue [g, i]
g i,
g, h i, k, (Cajori,
1928).
,
1812, . Cauchy
h kg i
, ,
. , Binet,
, resultante,
Laplace (, 1989).
Caley, 1841, :
'''
, a , '' a
a,
'''''' : , -, -+-+-,
. (Cajori, 1928).
,
.
, , ,
,
, .,
.
108
4
, ,
.
, ,
, .
,
, ,
.
,
,
(, 2001).
,
, ,
, .
,
,
,
, .
,
,
.
,
, ,
(, 2000).
109
4.1
,
,
, , ,
,
.
,
,
, ,
.
,
,
, .
,
,
. Bruner,
, ,
,
. : )
, , )
, ,
) ,
, (abstract)
(, 2000, , 1999).
,
. ,
,
. ,
,
.
(, , , )
110
,
(, 2000).
,
, ,
,
. ,
,
, ,
,
,
(, , ,
, ) ,
( , 1998, Hiebert, 1997, NCTM, 2004).
111
4.2
,
, ,
,
. , ,
, .
(symbolic relational
structures),
. ,
,
- - (Steinberg, 2000).
,
) - (sign/symbol) )
-, (object/reference, context) ) (concept)
Steinberg (2000),
.
,
(reciprocal)
. ,
,
(balance),
.
.
, ,
,
. Steinberg
(2000, . 141),
: (conveyance, signifier) ()
112
, (information, signified)
().
,
.
.
,
,
.
,
, .
,
(process) (concept).
, ,
(action) (entity),
.
.
,
(Gray & Tall,
1994, 2000, Hiebert & Lefevre, 1986 Gray & Tall, 1994).
, ,
- ,
, ,
. Gray & Tall, (2000),
,
, (interiorisation),
(encapsulation), (reification) ..
113
-
, Tall (2004),
,
. ,
(embodied), (proceptual) -
(formal-axiomatic) . ,
, .
. ,
.
,
,
,
.
Gray & Tall (1994, 2000),
procept,
(process) (concept)
. -
(elementary procept), :
.
- (procept), -
(elementary procepts) .
, -,
,
.
, ,
.
, ,
114
,
, .
,
(Tall, et al, 2001).
,
Gray & Tall (1994, . 1), : The ambiguity of
notation allows the successful thinker the flexibility in thought to move between the
process to carry out a mathematical task and the concept to be mentally manipulated
as part of a wider mental schema. ,
, ,
.
Sfard (1991),
,
(notion) , (operational) (structural).
. ,
(. 4).
, (deep
ontological gap) (. 4). , ,
, . , ,
,
. , ,
,
.
,
115
.
.
.
Sfard (1991)
: () (interiorisation) , ()
(condensation)
() (reification),
(operational) ,
, (structural) ,
.
Sfard, (1992)
.
,
.
,
.
,
,
. ,
. ,
, ,
,
. ,
,
,
.
,
. ,
, ,
.
, ,
116
(Sfard,
1995, 1992)
, ,
,
, ,
,
Vosniadou & Verschaffel, (2004), . ,
, ,
. ,
, ,
. ,
,
, .
, ,
,
(Vlassis,
2004).
,
Arcavi (1994, 2005),
, (symbol sense),
.
(behavior), ,
,
.
117
() ,
, ,
,
() ,
,
,
()
. ,
,
()
,
.
,
()
.
, Arcavi (1994) ,
.
,
,
.
118
4.3
,
, .
,
(it is subtle and difficult to
learn) (Davis & Hersh, 1981, Schoenfeld & Arcavi, 1988).
.
. ,
,
.
,
. ,
,
(Hughes, 1992,
1996).
,
. ,
.
,
(Schoenfeld & Arcavi, 1988).
-
,
. ,
. ,
119
(Tall, 1994).
, ,
. ,
,
, -
,
. , Vlassis
(2004), (structural signifier),
-
(operational signifier),
. ,
.
- Vlassis
(2004) :
() , , ,
, ,
,
() , ,
, ,
() , ,
.
, ,
, ,
. ,
-,
.
120
,
-
(Vlassis, 2004).
, Sfard (1991, 1992,
1994), ,
, .
,
,
,
. ,
,
,
, .
, ,
,
. ,
,
(Sfard,
1992).
,
, .
,
,
,
, .
, ,
,
121
,
(Sfard, 1994, 1995).
Sfard (1991), ,
,
( 5).
(Bourbaki, 1934)
(Skemp, 1971)
Property of a set or the class of all sets
of the same finite cardinality
0
1
5. (Sfard, 1991).
Sfard (1992),
, ,
, . ,
. : ()
() ,
. , Sfard (2000),
Ford (2003),
,
. ,
,
.
122
,
,
. ,
( ) .
,
, ,
. , ,
,
, Hilbert (Sfard, 1992).
-
,
, ( )
,
, .
, Tall et al. (2001),
( 6).
(process) (concept)
4
3+2
-3 3
3/4 ,
3+2x
v=s/t
y=f(x)
dy/dx
f(x)dx
6. (Tall et al., 2001).
123
, ,
,
, ,
(Tall et al., 2001).
, ,
.
,
, , . ,
, ,
, .
,
. ,
,
,
(proceptual thinking) (Tall et al.,2001,
Gray & Tall 1994).
Arcavi (1994, 2005),
,
,
, .
, ,
, .
,
-symbol sense
- Informal sense-making in formal
mathematics.
()
10% 10%;,
124
, ,
, ,
0,99.
() , () x2 y2=0
() (x-a)2+y2=1 0, 1, 2, 3, 4, 5, 6,7 8 ;, ,
,
.
() 3x+5=4x,
,
2412
++
xx =2
(4x+2)=2(2x+1),
,
, .
()
-1, , +1 , +1, +2 -2, -1, ,
,
,
.
() , y=b
, x=0 m=0 y=mx+b,
,
.
, , Arcavi (1994)
,
,
,
. , ,
,
,
125
. , Arcavi (2005), Sfard
(2000)
,
.
(serious tramp)
, .
, ,
Arcavi (2005), ,
,
,
,
.
126
4.4
.
,
,
(NCTM, 2004). (1989),
,
,
,
, ,
.
,
, - ,
, ,
,
, .
,
, ,
, . Sfard
(1995), ,
. ,
, ,
. ,
, ,
.
, ,
,
Recommended