Atque hoc momentum per plani
magnitudinem, puta per
, divisum; exhibet plani distantiam Centri gravitatis ab
,
: adeoque distantiam ejusdem
,
.
Hæc itaque à
distantia, in
(plani magnitudinem) ducta; exhibet
ejusdem
is momentum respectu
; feu Ungulam eidem
insistentem, cujus acies sit
.
Hæc denique Ungula (cujus altitude, in
, nulla sit, sed, in
, ipsi
æqualis:) fi ex planis ipsi
parallelis conflari intelligitur; e unt ea,
,
, & sic deinceps; hoc est, aggregatum omnium
,
, usque ad
.
Sunt autem ea plum (ut ex Gregaoii de Sanctio Vincentio, aliorumque post illum, doctrina constat) tanquam Logarithmi Arithmetice proportionalium
,
, &c. usque ad
; (feu
,
,
, &c. usque ad). Ergo Ungula ipsa, est eorundem Aggregatum. Hoc est (posito
,)
. Quod ostendendum erat.
Porro; cum sit
&c. (Quod dividendo
per
, patebit;) vel, posito
, (quó ipsius
potestates omnes deleantur,)
&c. seu
, &. in
. & similiter
in
& similiter in reliquis:
Erunt omnes , (spatium complectentes,) |
![{\displaystyle \scriptstyle {\left\{{\begin{matrix}\ \\\\\ \\\ \\\ \ \end{matrix}}\right.}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6bbeb1c61a753e5ef67a4a095e51389c844bbd6c) |
![{\displaystyle 1+a}](https://wikimedia.org/api/rest_v1/media/math/render/svg/998c74ac292731e915ab6f448f008194c107c4a4) |
![{\displaystyle {}+a^{2}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f2bd86aefdf69bd527a324f89c1f8cfb42210734) |
![{\displaystyle {}+a^{3}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ad1f7de293d2e08c9863481fff75399ac41fa18e) |
&c. |
|
![{\displaystyle 1+2a}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4da72e11f04a59355ebcf3b502bcbe006442603a) |
![{\displaystyle {}+4a^{2}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/623a3baf67fc60b1b39751047690245140992af6) |
![{\displaystyle {}+8a^{3}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b83a88e43ca1fce55b3d4ab0eacda151105a845d) |
&c.
|
![{\displaystyle 1+3a}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b6c6cb6be6b01c69e2c68fedc770884dde06c9f9) |
![{\displaystyle {}+9a^{2}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/767839845741b7117d71355069d316c4e12e443e) |
![{\displaystyle {}+27a^{3}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6f2ecfd63ab70346b4af6501c56b9e160b543acd) |
&c. |
in .
|
Adeoq; (per Arithm. Infin. prop. 64.) omnium aggregatum, seu ipsum spatium, erit —————— |
& sic deinceps usque ad
|
![{\displaystyle 1+A}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1206b21a6cca7d718eb46910bd0e45a0107a9fa7) |
![{\displaystyle {}+A^{2}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c5f7d30ce066cd9ca0e572a84ff7118f99af840b) |
![{\displaystyle {}+A^{3}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3725c6e5a9be2b599308384455abe3a9b81311ba) |
&c.
|
|
![{\displaystyle A+{\frac {1}{2}}A^{2}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/bcf032928d2d50ca871af2172f3e0ea29633a06d) |
![{\displaystyle {}+{\frac {1}{3}}A^{3}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2935acf3317444a0c5c85f7122a23e4141ec176b) |
![{\displaystyle {}+{\frac {1}{4}}A^{4}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/702118f719961f1e0f54a8ff37b45af1a2491409) |
&c, in .
|
|
|
Qualium Quadrato vel Rhombo
|
Ideoque, Plani
momentum respectu
; seu semiquadrantalis Ungula eidem insistens cujus acies sit
; seu planorum aggregatum ipsam constituentium; seu Logarithmorum summa quos ea representant,
,
in
: