y

Dh MotuCorpo-rum.

r8s Philosophia

37 6 . Lemma. Si fiiper hyperbolx EBDFasymptoto C N sumantur quatuor partesCG, C H, CK, CL, ut sit C G : C H^CK:C L; ducantur autem rectx G F,HO, KB, LE alteri asyinptoto CP pa-rallelae, & hyperbolae occurrentes in punc-tis F, D, B, E, juuganturque semidiame-triCF, CD, CB, CE, sectores hyper-boliciCBE, C D F erunt aequales. Agan-tur enim recta: BD, EF alymptotis oc-currentes in punctis M, O, N, P, & obparallelas KB, HD, CO erit MB:MKn D O: C H, & ob parallelas L E,G F, C P erit etiam N E: N L = F P:C G ; sed , ex natura hyperbolae interasymptotos ( Lern. I. de Conic. pag.115.) MB = DO, &NE = FP, unde M K= CH & NLzCG; Porrö CG:CH= CK;CL (per byp. ) hoc est, N L:MK = CK:CL = LE:KB, ex naturihyperbolae intrat asymptotos (Theor. IV.de Hyp. p. 114.) rectae igitur NE, MB,hoc eu, EF, B D erunt parallelx, acproinde, linea per earum medium X, Zducta erit Diameter, transibitque per cen-trum C; (Lern. IV. de Conic. p. 119.)unde facile deducitur trapezia MXZN,OXZP fore aequalia ut & areae mixti-lineae BXZE, DXZF, unde singulis excorrespondenti trapezio substractis relin-quentur area: MBEN & ODf P xqua-les, quibus addantur Triangula MBC,ODC, aequalia ob bases aequales MB»O D in eadem linea positas, & ob ver-tices ad idem punctum C concurrentes,erunt aequales arex CMNEBC, COPFDC,ex quibus denique substractis TriangulisNEC, P F C quae aequalia sunt ob basesaequales NE, P F in eadem linea posi-tas , & ob vertices ad idem punctum Cconcurrentes, supererunt sectores hyper-bolici C B E, C D F inter se aequales.Q. e. D.

377. Lemma. Si per puncta quaevis afymp-toti CL, agantur dux rectae GF, HDalteri asymptoto CP parallelx, & hyper-bolx occurrentes in F & D, jungantur-que semidiametri C F, CD, trapeziumhyperbolicum G F D H xquatur sectoriC F D. Nam, ex natura hyperbolae interalytnptotos, triangula CHD, CGF, x-quantur ob aequales angulos G & H &latera reciproca ( per Theor. IV. de Hyp.p. 114. ) adeoque sublato communi trian-gulo C G A, residua spatia G A D H, CAF

Naturalis

erunt xqualia, quibus si addatur idem spa-tium hyperbolicum DAF, summa: GFDH,C F D erunt xquales. Q. e. D. .

378. Coroll. r. Hinc iisdem positis qux( num. 376. ) trapezia hyperbolica GFDH,K B E L sunt xqualia.

379. Coroll. i. Si afymptoti partesCG, C H , C K fuerint continue propor-tionales , duo sectores C F D , C D B& duo trapezia hyperbolica GFDH,H D B K', xquantur. Eadem enim rationequi num. 37S. ostendetur rectam ITF tan-genti per punctum D ductx esse parallelam. •Unde si super asymptoto C L sumantur par-tes quotcumque CG, CH , CK, CL&c. in continua progressione geometrica,& ex punctis G, H,»K, L &c. aganturrectx GF, HD, KB, LE &c. alteriasymptoto parallelx, trapezia hyperboli-ca GFDH, H D B K , K B E L erunt ae-qualia j & vicissim si trapezia illa xquan-tur, erunt rectx CG, CH, CK, CL&c. in continua progressione geometrica.

380.

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