12 (Euclid (Eukleides : EÎkleÐdhc) : 300 ) (StoiqeÐwsic) ( ) 2 ( ) 2 16 3 17 18 (Introductio in Analysin Innitorum : 1748 ) 120 1
(1) (François Viète : 15401603) 1593 (Eectionum Geometricarum Canonica Recensio) 2 ( 1 p.372 pp.377-378) 3 A D BAC CD CE DE BC F B A F C BF F D F C ( 3) E AB = r AF = x BF = r + x F D = r 2 x 2 F C = r x BF F C = F D 2 BF : F D = F D : F C CD CE BC DE ADF 3 D F D 3 GF GF F D GF A B C G A F A AD AG AF B C F D BF F C AF AG AC AB BF F C AB AC AG AF BG F C GF BF BG F C ( 12) 3 x : y = y : x = d F D = y ( ) 2 d r = y 2 + x = BF = r + d 2 2 = ( ) 2 d y 2 + + d 2 2 = F C = r d ( ) 2 d 2 = y 2 + d 2 2 x = y 2 x 2
(2) (René Descartes : 15961650) 1637 ( ) (Discours de la méthode. Pour bien conduire sa raison, & chercher la verité dans les sciences) 3 (La Dioptrique) (Les Meteores) (La Géomètrie) 3 ( 1 ) ( 2 pp.35) 4 5 2 2 4 2 4 1 2 AB BD BC A C CA DE BE ( ) BE BD E D DE AC BC I E C D A B F G K H GH FG FH K K FIH G FH I GI ( ) 3
BD GH a b a + b a b a b ab a b a b a aa a2 a a 3 a 2 + b 2 a 2 + b 2 a 3 b 3 + abb C.a 3 b 3 + abb a 2, b 3 a 3 C.a3 b 3 + abb abb b 3 aabb b aabb 1 b 2 AB 1 AB 1 GH a BD b ( ) 1 ( ) ( ) a 2 a 3 4
x y ( ) 2 2 ( 2 pp.1820) 1 2 1 2 1 2 3 4 2 5 6 3 GL CNKL KN C EC CNKL KL BA GL G L CNKL AB EC AB A 1 C 1 C GA CB CB BA y x GA a KL b GA NL c NL LK c b CB y BK BK b c y G N C E K L B A 5
BL b c y b AL x + b y b CB LB c y b c y c a GA LA x + b y b 2 c 3 ab c y ab 1 xy + b yy by c yy cy c xy + ay ac b EC 1 CNKL CNK 1 GL EC 2 CNK L 1 KB (PĹppoc (Pappos) : 320 )) 5 2 1 3 4 2 5 6 3 y 2 = cy c xy + ay ac b C AB = x BC = y A ( AK x AG y ) C (x y) 2 2 y C F H I D G A B H E I x F 2 AB CDE CDE E D AB CD C HCI HCI (y a) 2 (x 2 +y 2 ) = b 2 y 2 DE = a CD = b 6
1 ( 2 p.5) 2 1 ( 2 p.10) (>Apollÿnioc (Apollonius) : 262 200?) 3 4 1 1 3 2 3 4 2 5 3 2 ( 2 pp.1213) AB AD EF GH C 1 CBG CDA CFE CHG CB CD CF CH 1 7
T S R E A B G H F C D AB CB AB A B x BC y 2 (2 ) AB A E G BC R S T ARB AB BR b AB x RB bx B C R CR y + bx R C B CR y bx C B R CR y + bx DRC CR CD c CR y + bx CD cy + bcx AB AD EF A E k EB k + x B E A k x E A B k + x ESB BE BS dk + dx y + dk + dx d BS CS y dk dx S B C C B S y + dk + dx FSC CS CF e CF ey + dek + dex AG l BG l x BGT BG BT fl gx y + fl fx f BT CT TCH TC CH g +gy + fgl fgx CH C 3 8
(3) (Pierre de Fermat : 16011665) (Ad locos planos et solidos isagoge : 1629 ) ( 6 p.128) 2 1 ( ) 1 (locus linearis) 2 2 1 2 ( 6 pp.128129) NZM N I NZ x NZI ZI y y dx = by I b : d = x : y x y NIZ N x Z M INZ N NZ NI 1 1 2 2 ( 6 pp.129130) R I O y N x Z M 2 xy = k 2 I NR ZI NZ M MO ZI NMO ( ) k 2 O NR NM 9
xy NZI NMO I x y xy d 2 + xy = rx + sy k 2 + xy = rx + sy X = x s Y = r y XY = k 2 rs x 2 = y 2 ( y = x ) x 2 = dy y 2 = dx b 2 x 2 = dy b 2 + x 2 = dy b 2 2dx x 2 = y 2 + 2ry b 2 x 2 = ky 2 x 2 + b 2 = ky 2 (x 2 + b 2 ) : y 2 x 2 + b 2 = ky 2 NO ZI b 2 : NR 2 O R N M R R RO N NZ x MR RO RO 2 OI 2 NR 2 : b 2 I y Z MR RO+RO 2 = RO (MR+RO) = RO MO (MR RO + RO 2 ) : OI 2 = NR 2 : b 2 MO OR : OI 2 = NR 2 : b 2 (MO OR + NR 2 ) : (OI 2 + b 2 ) = NR 2 : b 2 MN = NR MO = MR + RO = NR + NR + RO = NR + NO MO OR + NR 2 = (NR + NO) OR + NR 2 = NR OR + NO OR + NR NR = NR (OR + NR) + NO OR = NR NO + NO OR = NO (NR + OR) = NO NO = NO 2 = ZI 2 = y 2 OI 2 + b 2 = NZ 2 + b 2 = x 2 + b 2 y 2 : (x 2 + b 2 ) = NR 2 : b 2 NR 2 : b 2 (OI 2 + b 2 ) : (MO OR + NR 2 ) (x 2 + b 2 ) : y 2 I 10
( 3 pp.102103) 1 1 1 V 2 N M R IN IM IN IM INM O I NM = b ZI e N Z M NZ a 2a 2 + b 2 2ba + 2e 2 be NM Z 2 2 Z ZV ZV 4 NM VZ VOZ ZO ZM VO V VO OIR R RN RM RN RM RNM NM = b ZI = e NZ = MZ = a IN 2 = a 2 + e 2 IM 2 = (b a) 2 + e 2 (2a 2 + b 2 2ba + 2e 2 ) : be k : 1 4ZV : NM = k : 1 ZV = 1 bk ( ) 4 2 1 VO 2 = VZ 2 ZO 2 = 4 bk a 2 = 1 16 b2 k 2 a 2 V VO NM x ZI y ( x 2 + y 1 ) 2 4 bk = 1 16 b2 k 2 a 2 R (x y) x 2 + y 2 1 2 bky + 1 16 b2 k 2 = 1 16 b2 k 2 a 2 x 2 + y 2 = 1 2 bky a2 11
RN 2 + RM 2 = { (x a) 2 + y 2} + { (x + a) 2 + y 2} = 2x 2 + 2y 2 + 2a 2 ( ) 1 RN 2 + RM 2 = 2 bky a2 + 2a 2 = bky 2 RNM by ( RN 2 + RM 2) : RNM = bky : by = k : 1 ( 6 p.130) 2 (Tìpoi âpðpedoi : De Locis Planis) 2 (Apolloni Pergaei libri duo de locis planis restituti) 2 2 2 2 (Conics) 1 11 12 13 P J L X O B Z R E A M N H D K S G LZ ZH ZJ ( 8 pp.316317 7 p.53) latus rectum ( ) M DE ( BG) NM (linea ordinatim applicata : linea ordinata) ZH ZN (linea abscissa ) 2 2 (lineae coordinatae) (Gottfried Wilhelm Leibni : 16461716) 1692 12
(4) (Leonhard Euler : 17071783) x A AP ( 5 p.1) x y P PM ( 2 1 ) 1 RS 2 x RS x RS P A AP x = 0 P A AP x AP x RS A P AP x = AP P A ( 5 p.2 p.3 p.5 p.6) 4 x x y x x y x RAS x AP y PM AP B M R p P E A P D S m M 6 x y 12 AP x AP = x y PM = y 13
y y x 14 2 x y x A y RS x AB y y x x y = 0 ( 5 p.37) 67 x y y = 0 y = 0 x x x ( ) 1 0 = α + βx + γy 2 0 = α + βx + γy + δxx + εxy + ζyy 1 F. Viète(transl. by T. R. Witmer) The Analytic Art Dover 2006 2 R. ( ) ( 1 ) 1973 ( 48) 3 P. Tannery and C. Henry(ed.) uvres de Fermat Gauthier-Villars et Fils 1891 4 D. E. Smith A Source Book in Mathematics Dover 1959 5 L. ( ) 2005 ( 17) 6 ( 236) 1971 ( 46) 7 1980 ( 55) 8 I. Thomas(transl.) Greek Mathematical Works II Harvard U. P.(Loeb Classical Library) 2005 (1941) 9 1999 ( 11) 10 ( 18) 1978 ( 53) 11 2 2004 ( 16) 14