_Invenire gravitatem corporis longinqui ad circumferentiam circuli
ex particulis materiæ in duplicatâ ratione distantiarum inversè
attrahentibus constantem._
ESTO NIK (_Vid._ TAB. xxxiii. _Fig._ 1.) circumferentia circuli,
in cujus puncta omnia gravitet corpus longinquum S locatum extra
planum circuli. In hoc planum agatur linea perpendicularis SH, et per
circuli centrum X ducatur recta HXK secans circulum in I et K, et SR
parallela ad HX: producatur autem SH ad distantiam datam SD, et agantur
rectæ DC, XC, ipsis HX, SD, parallelæ. Tum ductâ chordâ quavis MN ad
diametrum IK normali eamque secante in L, ex punctis M, N, demittantur
in SR perpendiculares MR, NR, concurrentes in R; junctisque SM, SN,
erit SM = SN, MR = NR, SR = HL. Dicantur jam SD, _k_; HX sive DC,
_h_; XL, _x_; CX, _z_; XI, _r_; eritque HL = _h_ - _x_, et SH = _k_
- _z_. Est autem SM ad SH ut attractio (1 ⁄ (SM)²) corporis S versus
particulam M in directione SM ad ejusdem corporis attractionem in
directione SH, quæ proinde erit SH ⁄ (SM)³: sed est SR = HL, et (SM)²
= (SR)² + (MR)² = (SR)² + (SH)² + (ML)²; unde sit SH ⁄ (SM)³ = SH ⁄
((HL)² + (SH)² + (ML)²⁽³⁄²⁾), et ductâ _mn_ parallelâ ad MN, vis qua
corpus S attrahitur ad arcus quàm minimos M_m_, N_n_, exponitur per
(SH × 2M_m_) ⁄ (SM)³ = SH × 2M_m_ × ((HL)² + (SH)² + (ML)²⁽⁻³⁄²⁾). Est
autem (HL)² + (SH)² + (ML)² = _kk_ - 2_kz_ + _zz_ + _hh_ - 2_hx_ +
_rr_, hincque ponendo _kk_ + _hh_ = _ll_, ((HL)² + (SH)² = (ML)²)⁽⁻³⁄²⁾
= (1 ⁄ _l_³) + (3_kz_ ⁄ _l_⁵) + (3_hx_ ⁄ _l_⁵) - (3_rr_ ⁄ 2_l_⁵) -
(3_zz_ ⁄ 2_l_⁵) + (15_kkzz_ ⁄ 2_l_⁷) + (15_khzx_ ⁄ 2_l_⁷) + (15_hhxx_
⁄ 2_l_⁷), neglectis terminis ulterioribus ob longinquitatem quam
supponimus corporis S. Quarè, si scribatur _d_ pro circumferentiâ IMKN,
gravitas corporis S ad totam illam circumferentiam secundum SH, sive
fluens fluxionis SH × 2M_m_ × ((HL)² + (SH)² + (ML)²)⁽⁻³⁄²⁾ evadit (_k_
- _z_) × _d_ in (1 ⁄ _l_³) + (3_kz_ ⁄ _l_⁵) - (3_rr_ ⁄ 2_l_⁵) - (3_zz_
⁄ 2_l_⁵) + (15 _kkzz_) ⁄ (2 _l_⁷) + (15 _hhrr_) ⁄ (4 _l_⁷). Simili
modo obtinebitur gravitas ejusdem corporis S secundum SR. _Q. E. I._