Newton, Isaac
,
Philosophia naturalis principia mathematica
,
1713
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note49
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LIBER
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PRIMUS.</
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<
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>Nam in recta
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MN
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detur punctum
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N,
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& ubi punctum mobile
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M
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incidit in immotum
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N,
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incidat punctum mobile
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D
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in immo
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tum
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P,
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Junge
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CN, BN,
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<
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<
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CP, BP,
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& a puncto
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<
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P
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age rectas
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PT, PR
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occurrentes ipſis
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BD,
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CD
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in
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T
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&
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R,
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& fa
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cientes angulum
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BPT
<
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<
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æqualem angulo dato
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<
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BNM,
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& angulum
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<
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CPR
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æqualem angu
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gulo dato
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type
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CNM.
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Cum
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ergo (ex Hypotheſi)
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æquales ſint anguli
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<
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MBD, NBP,
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ut &
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anguli
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MCD, NCP
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;
<
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aufer communes
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NBD
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<
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&
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NCD,
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& reſtabunt
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æquales
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type
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NBM
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&
<
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PBT,
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NCM
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&
<
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PCR:
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adeoque triangula
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NBM, PBT
<
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type
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ſimilia ſunt, ut
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& triangula
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type
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NCM, PCR.
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Quare
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PT
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eſt ad
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NM
<
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ut
<
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PB
<
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ad
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<
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NB,
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&
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PR
<
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ad
<
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NM
<
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type
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ut
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PC
<
emph.end
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ad
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type
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NC.
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Sunt autem puncta
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B, C, N, P
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immobilia. </
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<
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>Ergo
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PT
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&
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PR
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datam habent rationem ad
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NM,
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pro
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indeQ.E.D.tam rationem inter ſe; atque adeo, per Lemma xx,
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punctum
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D
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(perpetuus rectarum mobilium
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BT
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&
<
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CR
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concurſus)
<
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contingit ſectionem Conicam, per puncta
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B, C, P
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tranſeuntem.
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Q.E.D.
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</
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<
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>Et contra, ſi punctum mobile
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D
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contingat ſectionem Conicam
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/>
tranſeuntem per data puncta
<
emph
type
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B, C, A,
<
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& ſit angulus
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DBM
<
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ſemper
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æqualis angulo dato
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ABC,
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& angulus
<
emph
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DCM
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ſemper æqualis angu
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lo dato
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ACB,
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& ubi punctum
<
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type
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D
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incidit ſucceſſive in duo quævis ſe
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ctionis puncta immobilia
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type
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p, P,
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punctum mobile
<
emph
type
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M
<
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incidat ſucceſſive
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/>
in puncta duo immobilia
<
emph
type
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n, N:
<
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per eadem
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n, N
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agatur Recta
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n N,
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<
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& hæc erit Locus perpetuus puncti illius mobilis
<
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M.
<
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Nam, ſi fieri
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poteſt, verſetur punctum
<
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M
<
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in linea aliqua Curva. </
s
>
<
s
>Tanget ergo
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punctum
<
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type
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D
<
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ſectionem Conicam per puncta quinque
<
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B, CA, p, P,
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tranſeuntem, ubi punctum
<
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M
<
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perpetuo tangit lineam Curvam. </
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>
<
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>Sed
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& ex jam demonſtratis tanget etiam punctum
<
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D
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ſectionem CoNI
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cam per eadem quinque puncta
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B, C, A, p, P
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tranſeuntem, ubi pun-</
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