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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 772, page id 851387705:0792
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De Calculo Centri Cravitatis.
Cap. V
hocest, ZC R= F Z VR = 3 s (Der SF. PrOB. 15.) Adeoque, Omn. 2 5VR: = de R
+55VR= - 53R
ErsO, Omn. Z a R2 = = 5 R2 + a5V R: = - = e R3 + + a 2 R2 = - s R, Mo-
mentum Solidi ABB, respectu perpendicularis plani A r a. VE prius. Unde Centri
graritatis inde diltancia habebicur. Aeque huic momenco, si addatur momentum
Solidi bBa B, habebicur momencum Solidi AbBa, idem quod alia mechodo jam
invenimus.
Deinde; Si insellieatur Solidum illud inceorum A ray ita poticum, ue sic A X0.
planum in licu horizoncali, adeoque a rn planum, sicu ad horizonrem recto: Mo.
mencum Solidi cocius, Duplum erit momenti Ingulae Semicirculi ADa, aciem
habentis Aa, respectu rectae ra. PF timilicer in parcibus respectire Cumpris. Sunt
enim (uc prius dietum G A, C.) Sinsula Parallelepipeda Solidum hoc complencia,
ad respectiras Pyramides complenies illam. Ongulam, uc 3 ad r: Forumque pa-
rallelepipedorum rectis Bb inliltentium diltantiae Cenerorum gravicatis ab erecto
plano ran EBb = =h, respectire; Pyramidum vero a r a plano, Diltanciae CCntro-
rum graricatis, respective, ZV a == Zb. Cum isiur magnicudines sinc uc 3 ad 1;
e diltantix, uF = 2d 3, seu 2 ad 3; erit (Propter ZX 3 =:Z) NIOMENEOruM TaLO, UE
2ad 1. Hloc elt, Momencum Solidi A7 a (hoc sicu potici) respectu rarectae, du-
plum momenti Ongula Semicirculi ADa, aciem habencis A an relpectu recta 20:
tit porrionis Solidi Ab Ba momencum, duplum momene Onsulx correlpondencis
Ba A, (aciem habentis Aa,) respectu ra. Elt aurm (per SC, F. prop. 16.) On.
rula Semicirculi, aciem habentis R a, momentum respectu r a, Z Ra; Ingulaque
Sectoris Ba A, 5V R3- F 3 52 R2 = S= V R. ErSO Solidi A F a liC pOlici, moten-
cum respectu ra Z R+; e PortionIS ABBa ZV R3 F 35=R = 8s= VR. (OUIX
te
est summa, omnium Z sB2, eo spectancium; lumpeis a arichmerice proportionali-
bus: Sune enim magnicudines, lumma omnium sh: ec distancia Z h relpective.) P. a
que momenta per respectiras magnitudines, 2R3, ee V R2 F 3S= R, (Per SA.
dirita; exhibent Cenerorum gravicatis ab erecto plano r an, distantias: Nempe,
cocius Solidi, Z R; et portionis A B B a = X + 4 s R = 52 v
60
Fix ilso autem portionis ABBa momento; si eximatur momentum portionis
bBa B: habetur momentum Segmenti ABB.
Est aurem bBa B, (per SB.) vh X seu s P; diltantia Centri graviratis (per
DrOP. Z hujus,) EbB = =B: Ergo momentum (in hoc situ) respectu rectae ra
Vh2R = =s= BR, Ceu (propter Vh = 5 2, e B = 2R = V,) S2 R2 = =s2 VR
Hloc itaque subduetum ex Portionis Solidi ABBa MOentO, ZV R3 f 352R2 -
ES=V R (S praeced.) Relinquit ZV R3 = Zs R 2 f Z5 2V R, momentum Solidi ABB
in hoc licu) respectu rectae r0. Quod per magnitudinem, V R = = s R,
(SB.) diritum; extibet ejusdem Diltanciam Ceneri Gravicatis a ra u plano,
4V X - 25 R + 252V
CR: (eu 42R = 2Vb R + 2V1 4R = 2BR + 2vh 20 R + 20b
60R -30b
6R- 3h- 30
ER F Zh: Adeoque, d BB, 2R = = h = ZV: FE propterca momentum ejusdem
AbB Colidi (se potici) respectu bB, L0 R - 5 sv R. = 30 A - 35 R-
65 VR-
Ouod quidem momentum, est summa omnium planiorum a B. eo spectancium.
(hoc cst, omnium VR, per SO, Q. PrOD. 27. Tumpris v arichmetice prOPOrTIONa.
libus usque ad V eorum maximum,) in suas respectire a b B distancias ductorum
hoc eltus in V=v respectire: Hoc elt, Omnium vY R, minus Omn. V2 R. Adeo.
que, (Der PrOP. F. hujus, ) EV 3R = 3 Y3R = EY3 R, (eu 3V3 R: ut prius.
Idem per se habebuur, (sine ope porrionis AbBa.) Cum enim singula rectis
bB insiltencia plana Solidum complentia, aequalibus incerrallis ab invicem distira,
sinc aequalia iplis a. Bv = v R respectire, (ut dictum est:) Sineque in diltanciis a
vertice P A, iptis v respectire aequalibus; erie cujusque momenium respectu FA,
V= R; adeoque summa omnium (propter v arichimetice proporrionales, ulque ad
maximum,) 3V R. (per prop. T. hujus.) Scu, relicuta v minulcula, ZV3R.
Adeoque propter magnicudinem, Ev= R, (per SB.) Cencri graricacis d F A diltan-
ti2 =V: ErBO, d7a, 2 R - ZV, tCu ZR f ZB; UE prIUS: EC propterea ejusdem,
relpectu ram momencum v’ R = 5 u R, seu Su R7 + 302 b R, hoc elt 30 8
25 K