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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)
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De Calculo Centri Crravitatis.
Cup. V
Fis. 200, TLm MF FigurX ConrOlrenda tig. 202. aequalem CurrX FiguIEX Convolut
N L L~ 13. 200. Ouippe Demonstratio prius achibita in comparatis invicem
polloniona Parabola, e spirali Lrchimedea; in aliis non minus obrinet. Nempe.
in parcibus sntinite exSIuIS, Habenda esse vrrobique T Y P crilinea, pro Friangulis
rectangulis, e quidem (propter aequales urrobique angulos ad E) invicem Cimili-
bus; adeoque, propter aequales utrobique Z F, aequales item erunt F I; acque
hoc Temper. Aqualis itaque cota MF curra Hg. 202. LOt MI Y Y F HB. 200.
Et propterea; ex cognira vel figurae Spiralis magnicudine, vel Curva Spiralis
longitudine; similiter cognolcitus vel Magnicudo tigura cujus convolutione tu,
vel Longicudo livex. Pc vice verla.
Et aurem, ex conroluta Parabola, tic Spiralis Archimedea; Fic, ex Paraboloci-
dibus conroluris, tiunt Spirales alixe; quarum radii M P crescunt in rarione angu-
Jorum ANC T duplicata, criplicara, aliatre mulciplicata; vel subduplicata, suberi
plicata, alialre Tubmulciplicata; aur eriam ex his urcunque compolita.
Ouod eodem modo oltendi porest quo in Parabola id octentum est; Tumpris
Parabolocidibus idoneis, in quibus roctae M B, MIH, SiC diridanEuE UE MI B, V. P,
Ec. hoc eAt Z Y, Z Y, _C. snE ipSIS MP, MP, _c. proportionales, (quo sectOreS
Z. NI T. H3. 200. ex rectangulis Z uV. Y fis. 202. convoluris tacti, similes Cnt,)
Paraboloerdisque curra per ip(as Z Y, Z Y, rectas cranscar. (Quod aliis aliisque
modis, pro variis Parabolocidium generibus, taciendum exit.) HToc enim tacto.
ostendetur, ut prius in Parabola, lingula Rectangula, in respectiros Scctores,
iplorum dimidios, conrolucum iri: reliquaque qua hinc lequuLIE.
Cum vero alius occurrat modus expedicior, aliunde sumpris principiis, ideu
praeltandi; (neque loctori, credo, displicebir varietas;) rem sic asgrediemur; ca
methodo qua ad Trithm. Insin. prop. 36. ecc. utus Cum.
L18. 199. In Spirali Archimedea, MTTF, lig. 199. Tumpris Sectoribus Cimilibus; 2d.
eoque cum raciis M F, M P, _c. HOC eA MP, MP, Scc. cum angulis P NIY
PMT; dec. arichmerice proporcionalibus; ut 2, 2, 3, 4, dec. Frunt arcus PY,
PY, Cmpore in ratione qux ex radiorum MP, et angulorum M P P, rarionibus
COMPONITUE,) UE F, 43 5, 16, ecc. leries Sccundanorum; (nempe uno gradu al-
tior quam elt Ceries Radiorum MP, propter ansulos M PY, in racione leriei Pri-
manorum, seu arichmerice proportionalium:) adeoque, si in rectas expandi intet-
ligantur, crunt ut tocidem ordinarim applicatae in complemenio Parabolae, (axem
habente MIP; alcicudinem, ip( P F, non ut prius hujus dimidio, aequalem,) ut
pote in ratione duplicata axium MIP, MP: (quales Tune PT, P P, complenes
Semiparabolx complementum MIT F II Ag. 202. Sumpris BF, FHI, Fg. 202. Xqua-
libus iptIS MF, FOP, fie. 299.) _ propterea, limul omnes, (hoc est U FP
spiralis complementum ad conterminum sectorem,) aeguancur Prienti Rectansule
Parabola circum Ccripri; Hoc Cst, SM PXPF, SCu ZMHXMF, (per L. vel 6. hu
jus.) F Aque Concerminus Sector M P PM, (nepore ex arcubus arithmetice. pro
POrTIONalIBS ContlatUS,) = ZBI LX P F, (per T. hujus.) Frso, figura Spiralis
(mepere secloris reliduum) == 6M PXP F, (PTODtEF Z = 3 = 5;) hoc est, (PrOPICK
218. 2993. Requis aurem taetivet cur parabola MF B tig. 202. alcicudinem, jam Donam
eo LBL. ipliuS arcus contermini P F H8. F99. longicudini aqualem; quam prius teceranr
arcus P F Fg. 200. dimidiam: racio in prompeu est. Cum enim recinencibus Tuam
Tingulis lonsicudinem rectis Z Y, uur in axem curratis; dum, in rectis MB, con-
trahuntur ee in unioum in punctum; adeoque Parabolae magnicudo decresear:
necesse ct u, in HF, rectae E0 (ipsis Z L aequales) prorratianeur puta in ZY
quibus respondeane, in Sectore Spirali concermino, areus OO;) adeoque Com
plementi magniendo creleat. Fr propierea, dum complementum contideramus.
(juxta hanc hyporhetin,) Tacienda erit H F (complementi Semiparabolae seu Se
miparabolocidis balis) iph PO F semper aequalis, (quicunque fuerit Parabolae seu
Parabolocidis gradus,) crpote quae ex omnibus =Y, (hoc M, omnibus OO,
componitur. Dum vero Parabolam vel Parabolocidem liguram M F B (juxra
priorem hypothelin) consideramus; facienda cst. M B Parabolae seu parabolocidis
alticudo, (non quidem omnibus OO, hoc est ipsi P O F,) Sed omnibus Z T simul
sumpris aequalis: hoc M in Spirali Archimedea, (ProDuer iplas M X, adcoque
2
202.
5. 3:: x. 3,) ad Sectorem conterminum, ut 1, ad 3