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# Title: De Locis Solidis Secunda Divinatio Geometrica (shelfmark Leibn. Marg. 151)
# Folio 99, page id 898203430:0127
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ORDTHATIONNM CONC. LIMITTOVS LIB. L.
Fles CaHSC AD, CD in HI, N; quae producta sCCCE A P, C 2.
ad axem BO aquidcancer ductas in P, 2.
Fic quium, in singulis FgurIS, sic, ve supra adnotanimus, quadratum
AO aequale rectangulo sub DO axe crianguli circumleripti in lemirectum
latuS OI, erit DO ad OA, vel DI ad LFI, ve AO ad OI, ac idco
rectangulum (ub DI, IO aequatur rectangulo (ub AO, III, vel sUb PI.
ETI, Sed illi sub DF, IO ReGUatUT quadracum sumpex mediae YF, qua
re FI media est quoque proportionalis inter PI, TFI, ideoque punctum
E cadic inter P, d HI: paricerque punctum O inter N, e 2, cadere.
demonstratoitur.
Insuper, quoniam duo timul rectangula IPTI, PIHI aequalia lunc qua-
Deato IP, Ture quadrato AO, Cve rectangulo DOI (ips aequali demon-
Arato) live rectansulo DIO cum quadrato 1O, de rectangulum P ITI
requale octentum tuic rectansulo DTO, ergo reliquum reccangulum IPII
reliquo quadrato IO aequale est, sivC quadratO PA. OuuM verO IF SIt
OAcula media proporcionalis seometrica iner PL, TFI, iplique L L. (u,
ex constructione, aequalis, de in directum IO, erit reccangulum OP F
aequale 2 rectangulo I P FI, Sive quadrato P A, hoc est rectansuium a Lemma
C D F. ad quadratum P A habebic rationem aequalitatis, quam habet praecedens.
quoque idem rectangulum 9) P F: ad quadracum lemiordinate ex P in FYy.
perbola equilatera L. Y, ergo P A aequalis est ip(i lemiordinate ex P:
quare deseripta Fyperbola P T cransuc omnino per A terminum ordina-
ex in Parabola per datam originem O: pariterque huic opposita Hy per-
bola SOV transic per alterum terminum C ordinatae ciuldem; sed re-
Ccangulum P. IHI aequatur quadrato lemiaxis transuersi PF, ve iam de
monstratum fuic, erBO reCtA AF D Fyperbolam concingit b ad A, SCd b Convert.
37. Primi
ipta AD continsic quoque ad A Parabolam ABC, ex Hy potheli, Hy. Conic.
perbola vero tora est ad rnam partem cansentis AD, Parabola autem.
tota ad alteram, erso huiusmodi Sectiones in terminis tantum A, C, OF -
dinate per Orisinem LC MUEUO CONTINSIINE CONUCXO.
Fit quum, in prima figura, semirectum OI incidat, ex Hypocheli, in
verricem B. Cique IF. sumpra media proporcionalis inter OL, LD crie
quoque media incer OB, BD, led lae sunt equales (ob LangenLeM AD)
ergo, et BF verique iplarum OB, B D equalis crit, hoc est minimus
ramus OB, ad Parabolam; aequalis lemiordinare B F, vique ad Sectio-
NEI OPPOTICAI AF.
In lecunda vero, de terria, ex vertice B agatur continsens Parabo-
Jam, quae oppositas secet in R, S, laterales vero tangentes in Y, X. Et
quoniam rectangulum OTD aequatur quadrato medix I F perpendicula-
ricer politae super OD, punctum L erie ad Circuli peripheriam, cuius
diameter OD; sed OB, BD sune aequales, ergo iuncta B F alteri ipli
rum, nempe OB aequalis eric.
Praterea ad axem F. P FEyperbola A F ordinetur ex l recta IM.
Frit ergo, ob Erperbolam aequilateram, rectansulum OM L aequales
quadrato MR sive quadrato lB, ee communi addito quadiato IL, rC-
ctangulum OM E. cum quadeato F I, vel unicum quadratum IM. Vet
quadeit.
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