![{\displaystyle 1-a+a^{2}-a^{3}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ddedd87c4db372e7fc57ba9159a6c761958c3dda) |
|
![{\displaystyle 1-2a+4a^{2}-8a^{3}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/bbf043d96588f7b9f22da3fd67fe8466cfae2706) |
|
![{\displaystyle 1-3a+9^{2}-27a^{3}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/bdca18d722cdf5d189a9a419a338642e984d8c9d) |
|
& sic deinceps usque ad
|
![{\displaystyle 1-A+A^{2}-A^{3}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/24687f65f0d4b8957cf9a4ecd302c31db872d707) |
|
Cum itaque sint |
![{\displaystyle 1+{}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8c70c475ed1c9277408bfe2ea41bf44ff0c18022) |
![{\displaystyle 1}](https://wikimedia.org/api/rest_v1/media/math/render/svg/92d98b82a3778f043108d4e20960a9193df57cbf) |
(usque ad ultimum) |
|
![{\displaystyle a+{}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ed28c20853ffca1d640713eb9171020c7721ddad) |
![{\displaystyle 2a}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d325c24be7d760207674a169b078892bdd5cbc76) |
(usque ad ) |
|
![{\displaystyle a^{2}+{}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/aa53ab41af07222800bd0cb621f476596cb56b11) |
![{\displaystyle 4a^{2}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/cde2fe47d440172f73d705e61119bcc23716fb7a) |
(usque ad ) |
|
![{\displaystyle a^{3}+{}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/69a3c6413bbae8a0a653b90432afe95193856700) |
![{\displaystyle 8a^{3}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5b2a0a92083b2fdc030f51ba499e402e6fe5dbf8) |
(usque ad ) |
|
& sic deinceps:
|
|
(quod ostendit ille prop. 16, estque à me alibi demonstratum:) Recte colligit, prop. 17. Expositum spatium Hyperbolicum
, &c. Adeoque si (assignato, ipsi
, valore suo in numeris, ut res postulaverit,) distribuantur ih duas classes
,
,
, &c. (potestates affirmatæ,) &
,
, &c. (potestates negatæ;) harumque Aggregatum, ex Aggregato illarum, subducatur; Residuum erit ipsum
spatium Hyperbolicum.
Nequis autem operam lusum iri existimet,, propter Addendorum seriem in utraque classe infinitam; adeoque non absolvendam: Hinc incommodo medelam (tacitus) adhibet: ponendo
, vel
, 21, aliive fractioni decimali æqualem, adeoque minorem quam
: (Hoc est, sumpta
minore quam
.) Quo fit, ut postriores ipsius
potesiates tot gradibus infra Integrorum sedem descendant, ut merito negligi possint.
Exempli gratia; positis
, &
2i. erit
|
![{\displaystyle A}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3) |
|
![{\displaystyle {\frac {1}{3}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7093420fb1b77a06432a4e0d9eba91705cef6d02) |
![{\displaystyle A^{3}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5e14d57c5ce0ca1c2537fa7e4eae914657ea1687) |
![{\displaystyle {}=0,003087}](https://wikimedia.org/api/rest_v1/media/math/render/svg/396b160acd12354bdb8f30a7db9274f27021bf00) |
|
![{\displaystyle {\frac {1}{2}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a11cfb2fdb143693b1daf78fcb5c11a023cb1c55) |
![{\displaystyle A^{2}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0a6f30e4c77b5a5e1e49ed2592e144389eade5ba) |
|
![{\displaystyle {\frac {1}{5}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1b977656fee2fb2e66210e174d78e2a484f89e45) |
![{\displaystyle A^{5}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/679cb05a978c6083b406dc9d71cd751c5e3f95c8) |
![{\displaystyle {}=0,0000081682}](https://wikimedia.org/api/rest_v1/media/math/render/svg/402595338a3a01b59e830b1b78c84f02d610c5ea) |
|
![{\displaystyle {\frac {1}{4}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a2dfb63ee75ec084f2abb25d248bc151a2687508) |
![{\displaystyle A^{4}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7bd81c6324f0335844619f31990711c14bc66f52) |
|
![{\displaystyle {\frac {1}{7}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7488185acac9cec8180b1f6a1f3c13b72d59b395) |
![{\displaystyle A^{7}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/250eb160e03942830df96962e007e56b3bb2de46) |
![{\displaystyle {}=0,000002572}](https://wikimedia.org/api/rest_v1/media/math/render/svg/391d8a0c54e7729166586609b1c4937564c26c38) |
|
![{\displaystyle {\frac {1}{6}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ba1caf4c96d913f6aafa9da0634f069fa42e0290) |
![{\displaystyle A^{6}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8ceecc51ea73dbb13c358da27e3a41415d2b7423) |
|
![{\displaystyle {\frac {1}{9}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f5e41e551372333b5c0d7d525ecf3797d67f980a) |
![{\displaystyle A^{9}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1ecb38da9e84089a49399ca2ab8d59934070238a) |
![{\displaystyle {}=0,000000088}](https://wikimedia.org/api/rest_v1/media/math/render/svg/036afbd3f4c0b6c2c1db78f42e654b693e028b5f) |
|
![{\displaystyle {\frac {1}{8}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b3f20dc5ae5815ab8628a1294c40639574e0c88e) |
![{\displaystyle A^{8}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b999139e99075630c07b3ab480993cf72e9de880) |
|
![{\displaystyle {\frac {1}{11}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4560a9297eaec321efc72830aa8688d241333737) |
![{\displaystyle A^{11}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1b2edca2bbfe4b8638c99077b85a46543c2d06f0) |
![{\displaystyle {}=0,000000003}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6320a2add38078b7d93b5ae00176f4d06986a615) |
|
![{\displaystyle {\frac {1}{10}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/cb7319d1751af9cc0419f3fba7803a4474f2bddf) |
![{\displaystyle A^{10}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/547c0552f1f05373c1725eb92ca175c5f769ea6f) |
|
|
![{\displaystyle {}+{}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/90db1521fed01f76ff6c513df8eabd4ce4127676) |
——— |
|
|
|
![{\displaystyle 0,022550984}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e3aeb810103c99e200772cc01cf4b356d00d78e7) |
|
Quæ est brevis Synopsis Quadraturæ suæ satis elegans.
Dissimulandum interim non est; sequis totius
spatii (cujus latus
longius intelligatur quam
) quadraturam postulet; rem non ita feliciter successuram: propter medelam, quam modo diximus, malo minus sufficientem. Cum enim jam ponenda
; manifestum est, posteriores ipsius potestates, altius in Integrorum sedes penetraturas, adeoque minime negligendas.
Huic autem incommodo, levi constructionis immutatione, facile subvenitur. Vid. Fig. 1.
Cæteris utique ut prius constructis; Quadrandum exponatur
ur spa-