Newton, Isaac
,
Philosophia naturalis principia mathematica
,
1713
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DE MOTU
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CORPORUM</
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SECTIO V.
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Inventio Orbium ubi umbilicus neuter datur.
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LEMMA XVII.
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Si a datæ Conicæ Sectionis puncto quovis
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P,
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ad Trapezii alicujus
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ABDC,
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in Conica illa ſectione inſcripti, latera quatuor infinite
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producta
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AB, CD, AC, DB,
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totidem rectæ
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PQ, PR, PS, PT
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in datis angulis ducantur, ſingulæ ad ſingula: rectangulum duc
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tarum ad oppoſita duo latera
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PQXPR,
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erit ad rectangulum duc
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tarum ad alia duo latera oppoſita
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PSXPT
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in data ratione.
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Cas.
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1. Ponamus primo lineas ad
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oppoſita latera ductas parallelas eſ
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ſe alterutri reliquorum laterum,
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puta
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PQ
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&
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PR
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lateri
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AC,
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&
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PS
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ac
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PT
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lateri
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AB.
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SintQ.E.I.ſuper
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latera duo ex oppoſitis, puta
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AC
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&
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BD,
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ſibi invicem paralle
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la. </
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la illa latera erit una ex diametris
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Conicæ ſectionis, & biſecabit eti
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am
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<
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abbr
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Rq.
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Sit
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O
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punctum in quo
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RQ
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biſecatur, & erit
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PO
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ordinatim applicata ad diametrum illam. </
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Produc
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PO
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ad
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K
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ut ſit
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OK
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æqualis
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PO,
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& erit
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OK
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ordinatim
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applicata ad contrarias partes diametri. </
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<
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>Cum igitur puncta
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A, B,
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P
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&
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K
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ſint ad Conicam ſectionem, &
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PK
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ſecet
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AB
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in dato an
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gulo, erit (per Prop.17 & 18 Lib. </
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Apollonii
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) rectangu
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lum
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PQK
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ad rectangulum
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AQB
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in data ratione. </
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<
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>Sed
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QK
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&
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PR
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æquales ſunt, utpote æqualium
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OK, OP,
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&
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OQ, OR
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differentiæ,
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& inde etiam rectangula
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PQK
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&
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PQXPR
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æqualia ſunt; at
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que adeo rectangulum
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PQXPR
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eſt ad rectangulum
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type
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AQB,
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hoc
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eſt ad rectangulum
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PSXPT
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in data ratione.
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Q.E.D.
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