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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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exOp. X
FINEFORUM
PROP. A
Corollarium.
Tque in(uIper, Linea Spirales hae una duabus, cribus, quatuor, 33c
circulationibus factae, (puta M A, N AB, M ABC, M ABCD,
se habent ad invicem ut numerorum Prithmetice proportionalium quadrata;
nempe ut 2, 4, 9, 26, 3c. Sive sunt in duplicata ratione rectarum M A, MIB,
MTC, ATD, CC.
Sunc enim recta M A, AB, BC, CD, invicem aequales (proprer aequabilemm
progreltum puncti mobilis in recta MIA, etiam continuata, rantum in una circu-
latione aequabiliter progrediencis quantum in alcera;) adeoque tam radii M A,
MB, MC, ND, quam Peripherix (illis ractiis deseripea) A, B, C, D, FunE ad
invicem uC E, 2, 3, 4. Si igicur sumancur peripheriae, prima lemet, secunda bis,
texEIA TeT, quarra quater, crune illa mulcipla, (puta Y A, 2B, 3C, 4 D,) ut nu-
MerI quadrati 2, 4, 9, 26, (VC LXI, 2X2, 373, 4X4. RC propterea etiam hO-
rum mulciplorum lemilles, hoc est (per PTOP. 5.) Spirales M A, M AB, M ABC,
M ABCD.
Vel aliier, Si circuli primi peripheria ponacur A = p, erie secundi B = 2 p,
TCrEII C = 3 p, quarn D == 4 P, _ SiC deinceps; e IA = E p, 2B = 2X2 p == 4 p.
D = 4 X4 p = 26 p: _C. EL (per PrOD. 5.) Spirales
et a = 3 p, t A B = EB =3B, M ABC = 3C = 2p, MABCD =3B = 18 p, _c.
ideoque ad invicem UE F, 4, 9, 26, ecc. hoc est in duplicata ratione rectarum
MlA, MB, M Cy MD, eec. (quae sunc ad invicem HC 1, 2, 3, 4, ec.) Quod
erat demonstrandum.
E ROP. XI
Corollarium.
universaliter; Ejusdem (vel similium) Spiralis segmenta quaelibet
(a principio Spiratis exorsa) sunt ad invicem in duplicata ratione re-
ctarum conterminarum.
it
Cum enim (per constructionem linex spiralis) cadem siE racio rectariim M. I.
NI F, gyae est angulorum PM F, PM P, (anguli voce eo sensu sumpra qui
Tupra InNUITUI pTOP. 5.) ratIO arCuU PF, PF, (quae ex duabus illis rationibus
COMPONITE,) ut e spiralium MF, M F, (quae istorum arcuum sunE semilles,)
erit duplicata rationis rectarum M P, M T, Peu VE M F9, M F9
Sic V. 8. E recta M A (unius circulacionis) dicacur Fr, e peripheria circuli
primi (eo radio descripra) E P, eric spiralis M A = Z P. Pgitur, in una circula-
tione cum semisse, Aec recta contermina L Zr = Zr, e Decipheria concermini cir-
culi 3B, quae ducta in 3 (numerum circulationum) AE Z X 3 X p = Z p: Ejusque
semiltis - p = 8p est spiralis eodem tempore deseripra.
Similes autem appello Spirales, si recta M A, MB, MC, Bc. in unam aequa
les sint homoloeis rectis in alsera.
PROC. LLL.
Corollarium.
T vero ejusmodi spiratium assu (Palco S. LA B In una tanta sit
quanta VIC in alia) rectae conterminae sint aequales; erunt Spiralium il
larum sermenta homoloeis sant tenm (patio a Alala B M A in altera)
tel or 6b voblatted.
Nam