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# Title: [Johannis Wallis S. T. D. Geometriae Professoris Saviliani, in Celeberrima Academia Oxoniensi; Opera Mathematica] Johannis Wallis S. T. D. Geometriæ Professoris Saviliani, in Celeberrima Academia Oxoniensi; Opera Mathematica (shelfmarks Leibn. Marg. 100:1/2, Stück 1; Leibn. Marg. 100:1/2)
# Folio 344, page id 851387705:0364
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DE SECTLONEBUS PFOP. XXXVII.
Iua Synthetice sie demonstratur, (reperiris Analyleos veltisiis.) A centro C
ducatur ad Flyperbolam C. 4. Reque in puncto a Fyperbolam concingat a F, dia-
metro cnilibet FIA OcCurtenS in F; ipsique parallela quaelibet OO, eidem dix-
MetrO TIA occurrens in C, e rectae C a (productae) in n. Diameiro autem FI A
ordinatim axplicetur a HI, eique parallela nh, eidem diamerro occurrat in h. Fa.
demque quae prIUS MANCANE CymbOla B, d, L, 2,6, 8, 6, 7,5.
Sumantur autem in diametro verinque a punctoh aequales rectae hD = V   5 a 2
atque ordinatim pOnantur DO, DO, recta OO occurrentes in OO. Adeoque
bilecantur tam DD in l (ver construetionem,) quam OO (propier paralle.
1a5) in n.
Dico rectam OO (a recta Cian bisectam in n) Hyperbolae inscriptam este;
hoc est puncta O O ette in ipla Fyperbola curra. Quod sic probatur;
Ouoniam sune (ut in Analysi demonstrarimus) HF = =, 1C -
HA = d f =, e na = 2 f =: Erit DG (=hC F HD) -7 d = , ded2
icem D A ( = h A = nD) = a f 2C + y 33d2
62
D2 = 2 + a + 53d 2
E rectangulum D A  D a =  d 2 + 538 + 53d3 + 2 7C V53d2. (nempe pro
Q e q in analssi, substiuendo in synchesi 3a=  Y  3d =.) vel (quia 
=DAXDa.
Tum (propter, similia triansula) erit HF. Ha:: DG, DO. hoc est, ==. B::
Yd Z +, 3d 2 yd 2 = C Y SEd2 p = D O. ejusque quadeacum Y  2
+53d 3C +27d ZCY SEdE p = DO9
Nd2 = 52c2 = 2yCVdEd2
Sed de tancundem est quadratum Ordinatim applicatae puncto D. Ntam (pro.
pter Hyperbolam) uc rectangulum IIAX II a, ad rectangulum D A XD a; lic HI a 9
quadracum ordinarim axplicata puneto H, ad quaruim, quod erit quadrarum
ordinarim applicacae puncto D. noc cA a ∞, 3 a 2 + 53C = 2rCY53d2.: B.
stiis
trie
id
242 CC  2y ∞ N d S. d_V, JOTTLBIB, C
viess.) sbu
43.   7 6. quod aequale est ips DOq ut modo ollendimns.
EE PrOPIerea DO, DO, (quae, ex constructione, sunc punctis DD ordinatim
polita,) lune et ordinacim applicatis aequales, adeoque puncta OO sunt in FIyper.
bolX curra rectaque OO ett Eyperbolae inscripta. Sed et illa (eique parallelae,
eadem rarione,) a recta Can bilecatur; adeoque ordinacim inseripra est ejusque
Semistis nO drametro C. an Ordinatim applicata. Ouod crat demonstrandum.
Nocandum aurem elt (quod et prius monni) tieri quidem posse ue interipta
O0 diametrum cransear in C incra hyperbolam. Quoa TaMEM ODETACIONIS PIO.
prectum non alicer immucabir quam rariando nonnunquan signa f =, auc quan.
titatem negatire incerprecandam exhribendo. Or experienti patebit.
Recta autem illa a C. quam FEyperbola Diametrum esse ostendimus, iisdem op-
positis Erperbolis convenit quibus:  AC; ipsaque a C. duplicata erit earundent
Diameter Transversa. Sumatur, enim in oppolica Hyverbola ad = A HI, hoc est
Cd = CFI, d ordinarim applicerur (sieu Contrario) do, quae eric ipsi H a aequa.
lis de Parallela (proxrer aequales diamerros interceptas, candam cranlverlam, idem
lacus rectum, acque licum parallelum, per prop. 34.) adeoque anguli alterni
xquales, nempe Cido = CHa; _ propterea (juncta oC) criangula C. Ila, C d0,
similia et aequalia; ideoque (propier anxulos  ropau ad C aequales) CriE a C.0
una