DE MUNDI
SYSTEMATE
SYSTEMATE
LEMMA V.
Invenire lineam curvam generis Parabolici, quæ per data
quotcunque puncta tranſibit.
quotcunque puncta tranſibit.
Sunto puncta illa A, B, C, D, E, F,&c. & ab iiſdem ad rectam
quamvis poſitione datam HNdemitte perpendicula quotcunque
AH, BI, CK, DL, EM, FN.
quamvis poſitione datam HNdemitte perpendicula quotcunque
AH, BI, CK, DL, EM, FN.
Caſ.1. Si punctorum H, I, K, L, M, Næqualia ſunt inter
valla HI, IK, KL,&c. collige perpendiculorum AH, BI,
CK,&c. differentias primas b,2b,3b,4b,5b,&c. ſecundas c,2c,
3c,4c,&c. tertias d,2d,3d,&c. id eſt, ita ut ſit AH-BI=b,
BI-CK=2b, CK-DL=3b, DL+EM=4b,-EM+FN=5b,
227[Figure 227]
&c. dein b-2b=c,&c.
& ſic pergatur ad diffe
rentiam ultimam quæ hic
eſt f.Deinde erecta qua
cunque perpendiculari
RS,quæ fuerit ordina
tim applicata ad curvam
quæſitam: ut inveniatur
hujus longitudo, pone
intervalla HI, IK, KL,
LM,&c. unitates eſſe,
& dic AH=a,-HS=p,
1/2pin -IS=q, 1/3qin
+SK=r, 1/4rin +SL=s, 1/5sin +SM=t; pergendo videlicet
ad uſque penultimum perpendiculum ME,& præponendo ſigna
negativa terminis HS, IS,&c. qui jacent ad partes puncti Sver
ſus A,& ſigna affirmativa terminis SK, SL,&c. qui jacent
ad alteras partes puncti S.Et ſignis probe obſervatis, erit
RS=a+bp+cq+dr+es+ft,&c.
valla HI, IK, KL,&c. collige perpendiculorum AH, BI,
CK,&c. differentias primas b,2b,3b,4b,5b,&c. ſecundas c,2c,
3c,4c,&c. tertias d,2d,3d,&c. id eſt, ita ut ſit AH-BI=b,
BI-CK=2b, CK-DL=3b, DL+EM=4b,-EM+FN=5b,
227[Figure 227]&c. dein b-2b=c,&c.
& ſic pergatur ad diffe
rentiam ultimam quæ hic
eſt f.Deinde erecta qua
cunque perpendiculari
RS,quæ fuerit ordina
tim applicata ad curvam
quæſitam: ut inveniatur
hujus longitudo, pone
intervalla HI, IK, KL,
LM,&c. unitates eſſe,
& dic AH=a,-HS=p,
1/2pin -IS=q, 1/3qin
+SK=r, 1/4rin +SL=s, 1/5sin +SM=t; pergendo videlicet
ad uſque penultimum perpendiculum ME,& præponendo ſigna
negativa terminis HS, IS,&c. qui jacent ad partes puncti Sver
ſus A,& ſigna affirmativa terminis SK, SL,&c. qui jacent
ad alteras partes puncti S.Et ſignis probe obſervatis, erit
RS=a+bp+cq+dr+es+ft,&c.
Caſ.2. Quod ſi punctorum H, I, K, L,&c. inæqualia ſint inter
valla HI, IK,&c. collige perpendiculorum AH, BI, CK,&c.
differentias primas per intervalla perpendiculorum diviſas b,2b,
3b,4b,5b; ſecundas per intervalla bina diviſas c,2c,3c,4c,&c.
tertias per intervalla terna diviſas d,2d,3d,&c. quartas per
valla HI, IK,&c. collige perpendiculorum AH, BI, CK,&c.
differentias primas per intervalla perpendiculorum diviſas b,2b,
3b,4b,5b; ſecundas per intervalla bina diviſas c,2c,3c,4c,&c.
tertias per intervalla terna diviſas d,2d,3d,&c. quartas per
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