tium; (cujuscunque fuerit longitudinis
; puta major minorve quam
, vel huic æqualis: sumptoque ubivis inter
&
, puncto
; puta ultra citrave punctum
, vel in ipso
puncto:) Ponantur autem (non, ut prius
, &
: sed)
; &
quæ intelligatur in æquales partes innumeras dividi, quarum quælibet sit
. Erunt iraque, post
, reliquæ deinceps decrescentes
,
,
, &c. usque ad
. Item, propter æqualia Rectangula
,
,
, &c. puta,
: Erit
reliquæque deinceps
,
,
, &c, usque ad
spatium
complentes. (Quæ omnia ostensa sunt, in mea Arithmetica Infinitorum, prop. 88, 94, 95.)
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![{\displaystyle b^{2}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/acf98b04bfc723606ebb4a7942fa3ab94becd2ee) |
![{\displaystyle -b^{2}a}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e9379025f54633aec116697647f13ae9922bf730) |
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![{\displaystyle +b^{2}a}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3b8e38f9bc3819626fe18ddc1dd4b668262cd06a) |
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![{\displaystyle +b^{2}a^{2}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5b39dfe4f8eb77047e5b1e43d549b9a4b2e3880a) |
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![{\displaystyle +b^{2}a^{3}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c5dc63c48dcd7d60b182c772dd2e0c740b1f379c) |
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![{\displaystyle +}](https://wikimedia.org/api/rest_v1/media/math/render/svg/fe6ef363cd19902d1a7a71fb1c8b21e8ede52406) |
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Factaque Divisione; re perietur
, &c.
Hoc est,
in
, &c. (sumptis ipsius
potestatibus continue sequentibus affirmatis omnibus.) Cumque de reiquis idem sit judicium; erunt rectæ omnes, ipsis
&
interjectæ,
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![{\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. |
in ,
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![{\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. |
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![{\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. |
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& sic deinceps usque ad |
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![{\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. |
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Omniumq; Aggregatu, |
![{\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 . |
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(per Arithm. Infin. prop. 65.) |
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Exempli gratia.
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Positis ![{\displaystyle AH}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5ca358eadf4fc37b14a8c6f818ce4db6c07757c9) |
.
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![{\displaystyle Hr=A}](https://wikimedia.org/api/rest_v1/media/math/render/svg/81c914bd97e93071157915b447043ca1394e8de7) |
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![{\displaystyle AI=b}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9ffea21478e6eef7b2657980161ab943132dadd0) |
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Adeoque ![{\displaystyle b^{2}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/acf98b04bfc723606ebb4a7942fa3ab94becd2ee) |
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Erunt |
![{\displaystyle A}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3) |
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![{\displaystyle {\frac {1}{2}}A^{2}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/065994d4a33bb862bfff6ea43c0f71d18f9d45df) |
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![{\displaystyle {\frac {1}{3}}A^{3}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/fc3f74a803ceab1acb6e760ae188da71a53e812c) |
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![{\displaystyle {\frac {1}{4}}A^{4}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/21b7792d0712863028170f9dd439afb76d67d85a) |
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![{\displaystyle {\frac {1}{5}}A^{5}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4b11351938234f19169b2e3f27833522dff0bc82) |
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![{\displaystyle {\frac {1}{6}}A^{6}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7c521040391730b342914fbacc94d59627a65ad0) |
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![{\displaystyle {\frac {1}{7}}A^{7}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/582b0b63e49b221c741db2e72335cd5b6d51def9) |
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![{\displaystyle {\frac {1}{8}}A^{8}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9deaa2262e9ee4dbf1a8121d181055581b289989) |
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![{\displaystyle {\frac {1}{9}}A^{9}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/14562be4c11083c1b2417e7c436133851e40a02e) |
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![{\displaystyle {\frac {1}{10}}A^{10}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ddf99786f0c665640e779ad252a03109001ae03c) |
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![{\displaystyle {\frac {1}{11}}A^{11}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/02bb86a5d93e1b2acc6dc7692822662b43bec7c5) |
![{\displaystyle {}=0,000000003+{}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3d527bcfc6aadf562fccbfafee633e9f1072a14e) |
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Horum summa |
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Ducta in ![{\displaystyle b^{2}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/acf98b04bfc723606ebb4a7942fa3ab94becd2ee) |
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Exhibet —— |
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