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Lect. VII.
ADhuc in _Veſtibulo_ hæremus;
nec aliud quàm velitamus.
I.
Sint duo quanta A, B;
quorum majus A;
adſumpto tertio quo-
piam X, erit A + X. B + X & lt; A. B.
piam X, erit A + X. B + X & lt; A. B.
Nam ob X.
A &
lt;
X.
B;
erit componendo X + A.
A.
&
lt;
X +
B. B. vel permutando X + A. X + B & lt; A. B.
B. B. vel permutando X + A. X + B & lt; A. B.
II.
In linea YZ ſignentur tria puncta, L, M, N;
&
inter puncta
L, N ſumpto puncto quopiam E, alteróque G extra LN (verſus Z);
ſecetur EG in F, ut ſit GE. EF : : NL. LM; cadet punctum F ad
11Fig. 61. partes MZ:
L, N ſumpto puncto quopiam E, alteróque G extra LN (verſus Z);
ſecetur EG in F, ut ſit GE. EF : : NL. LM; cadet punctum F ad
11Fig. 61. partes MZ:
Nam eſt N E.
ME &
gt;
NL.
ML :
: GE.
FE &
gt;
NE.
PE.
22* I _hujus_.
ergo FE &
gt;
ME.
III.
Sint rectæ BA, DC parallelæ item rectæ BD, GP parallelæ;
pérque punctum B ducantur utcunque duæ rectæ BT, BS ipſam GP
33Fig. 62. ſecantes punctis L, K, dico fore D S. DT : : KG. LG.
pérque punctum B ducantur utcunque duæ rectæ BT, BS ipſam GP
33Fig. 62. ſecantes punctis L, K, dico fore D S. DT : : KG. LG.
Nam eſt KG.
LG = KG.
GB + GB.
LG = PK.
PS +
PT. PL = DB. DS + DT. DB = DT. DS.
PT. PL = DB. DS + DT. DB = DT. DS.
IV.
Eſto triangulum BDT, basíque DB parallelam quamvis PG
interſecent per B ductæ quæpiam duæ rectæ BS, BR punctis L, K;
dico fore LG x TD + KL x RD. KG x TD : : RD. SD.
interſecent per B ductæ quæpiam duæ rectæ BS, BR punctis L, K;
dico fore LG x TD + KL x RD. KG x TD : : RD. SD.
Sumantur enim BM = GP, &
BN = LP;
&
BO = KP;
44Fig. 63 unde conſtat junctas PM, PN, PO ipſis TB, SB, RB (reſpectivè)
parallelas eſſe. Et quoniam eſt DM. PD : : DB. TD. erit DM x
TD = PD x Db. Similiter eſt DN x SD = PD x DB. quare
DM x TD = DN x SD = DM x SD + MN x SD, tranſpo-
nendóque DM x TD - DM x SD = MN x SD. Simili
44Fig. 63 unde conſtat junctas PM, PN, PO ipſis TB, SB, RB (reſpectivè)
parallelas eſſe. Et quoniam eſt DM. PD : : DB. TD. erit DM x
TD = PD x Db. Similiter eſt DN x SD = PD x DB. quare
DM x TD = DN x SD = DM x SD + MN x SD, tranſpo-
nendóque DM x TD - DM x SD = MN x SD. Simili