652 / 798
Catoptric~
éhque
ut
prius perpendiculari CD, junll:aque
li-
HG,non íecare hyperbolan1 in aliqtto P!tnél:o in·
nea llC,ducarur lmea CO,perpendicólaris ad llD,
ter H , & C, pofito verbi gracia in pmrll:o
!.
Ar-
dico l111eam CO, ita duci uc non attingac hyper–
bolam ultra punél:mn C veríus H, in nullo enim
puné\:o attingere pottll hyperbolarn,Nequc enim
atringec in punél:o H. Ducan:r enim linea HCI,
&
DH, icem pcrpendicularis HF,
&
llH.
Demonllrario. Gum augulus DFH
fo¡
ponatur
reél:us , (
per
1
7 .
i.)
erit lacus DH, rnajus lacere
!?
\":!.
Sunr aurem (
per cor.
1.
pr.cedemu)
lineo:
ingar enim fi fieri porel1: punéh1m 1,ducatur ra•
dius
BI,
perpendícularis
10,
&
linea IF.
Demonftratio. Cum in criangulis BIG, FIG,
angtlli in punél:o G fine rell:i, irem linea: DG,GF
:z:quales,
&
GI, communis (
per
4.
1. ) crunc·ba–
fes Bl,Fl :z:quales; íed jam
(ex defcrirtionepara–
rabol~)
linere BI, ID :z:qnales Íunc: igicur line:z:
ID, IF effcnt requalcs, quod abfurdum ell:, cum
angulus D rell:us !ir.
C O R O L L A R 1 U M.
FH,
llH :z:quales; igirur linea DH major eft re–
lh
BH. In criangulis aurem DCH , BCH, cum
latera llC, CD, (
per cor.
1. )
fint :z:qualia,larus
CH commune,& pafis DH major oftenfa
lit
b~fi
BH
(
per
•
5.
1.)
erir
angulu~
DCH, íeu
!CK ,
illi
ad verricem oppofirus , rnajor angulo llCH;
ha–
benr aurem communem angulum llCK , ignur
anguJus ICll, major eft angulo KCH,
&
i\11 op–
poliro !CD. In rtiangulis igicur ICD, IGB, cum
latera DC, llG linr :z:qualia;
&
larus CI commu–
nc, item angulus IGD minor angulo IC!l (
per
25.
1.) etit bafis
ID,
minor bafi ll I; quare per–
pendiculatis CO,qu:z: dividir bifariam lineam llD
cadet infra punél:um
l,&
poft punél:um
e,
cadet
fupra punél:um H ; ergo non fecabir hyperbolam
in
punél:o H. , Quod auccm demonlhavimus de
punél:o H,ollendam de quoliber alio punll:o; igi–
tur non arring'ct hyperbolam ultra punél:um
C.
Rcftu
nr
oftendam cam lineam non atringere
Sequimr ex hoc lcmmare lincam HG
dfe
ran-
hypcrbolam ince( punél:mn G, & punll:um A. Sir
gemern parabol:z:, cum nullibi attingar parabo-
enim hyperbola in
~ua
paricer punll:um A
lit
ver-
lam, nili in puné\:o H.
tex, Bfocus, lirque punél:um
H.
Duc~tur
parirer
'
pcrpendicularis HF,
&
radius BH , 'conjungacur
l1.!l!Z!illll!!ilflllflllllfl®'®llfl;l1.!l,lll.\ll!ll1.!lllill1.!l!1:!!!1.!ll1.!l~llli
linea llF ," fecans hyperbolam in puné\:o C ; dico
fi
a
punll:o
H,
ducamr perpendicularis ad BF,
hrec rota cader exrra parabolam.
. H:r:c enim perpendicularis HG,ur fupra ollen–
c!Jmus,dividir bifariam lineam llF, c!im autem li–
~e:z:
llG, CE lint :z:qualcs,
&
linea
CF,
!ir
tpajor
!mea CE.propter angulum rell:um E,erit CF,ma–
jor quam C ll ; ergo punébum in quo
13F
divide–
cur bifariam' erit ultra
e ;
ergo ¡am habemns
quod linea HG,uon atringit punél:um C fcd 'cadic
extra.
Neque eriam aninger ullnm punll:um imer
A
&
e
polimm ; attingar cnim
6
fieri poteft pun–
Ctum
1,
ducaturque perpendicularis 10,radius lll
&
linea IF.
'
Demonftmio.
ln
triangulis.BGI, FGI,cilm ;
111•
guli ad G, linr rcél:i, irem line:z: BG, GF requales,
&
IG, communis; crunt linere Bl,IF req1alcs;fed
l
D,
l
B,
funt
jam :z:quales
(
ex
de(criptione
h)•perbol~;)
ergo line:r: ID,
IF
etfenr :z:quales,quod
ell:
abíurdum, cum angulus D
!ir
reltus.
Denique affero lineam ex punél:o
H
dnél:am
l'~'pendiculariter
ad lineam
BF ,
fcilicct
lmeam
P R O P O SI T l O XX l. ·
Theorema.
Parabola
tmiHadios
º'"''"
axi
earallelos
Ín
¡nm[/o
foci.
Sir paraboJa ctijus vemx A, focus
B.
_axis
All;
incidar radius DC axi
AB
parallelus, d_ico
11l~m
rdlcél:endum in focum B. Ducatur ·en1m radms
BC, concinuemrquera·dius
OC,
íir~uc
AG, ipíi
AB requalis , ducarur perpendiculans CE.. ltcm
!mea BE ad quam
CJ!;
punél:o
e
ducarur perpen–
dicnlaris CF, qure
(per
lemma
fi•perius)
rangcr pa-
rab~~~~¿nll:ratio.
Radius incidensDCE 'cum lit
parallelus axi ·BA , qui perpendicularis ell: linc:r:
GE, perpendicularis ctiam•erir ad eandem
E·
Sunr ergo line:z: BC, CE :z:quales,
(per deflr.iptio–
nem
parabol~
fi•pra
traditafn.)
~rnre
in
rrian~u
lis
fEC ,
llFC, cum angllli ad
F
lim
~c~i;o,&
F<;'.
larus