Full text: Gravesande, Willem Jacob: Physices elementa mathematica, experimentis confirmata, sive introductio ad philosophiam Newtonianam

185. SCHOLIUM 5.
De Motu in Ellipſi agitatâ.

Corpus in Ellipſi retinetur vi centrali ad focum tendente & juxtarationem
inverſam quadrati diſtantiæ decreſcente , ſi ſuperaddatur vis quæ decre- ſcat in ratione inverſa cubi diſtantiæ, eandem corpus deſcribit Ellipſim trans-
latam ita, ut eandem partem verſus motus ipſius cum motu corporis diri-
gatur Vis ultima magis decreſcit, auctâ diſtantiâ, quam prima; idcirco ſum- ma virium, celerius decreſcit quam juxta rationem inverſam quadrati diſtan-
tiæ, unde conſtat propoſitio n. 386.

185.1.

423.
381. 411.
420. 421.

Simili demonſtratione conſtat n. 387. nam ſi ex vi quæ ſequitur rationem
inverſam quadrati diſtantiæ tollatur vis, quæ ſequatur rationem inverſam cubi
diſtantiæ, id eſt primâ celerius decreſcens, quæ ſupereſt lentius quàm juxta
rationem inverſam quadrati diſtantiæ, auctâ hac, minuitur.

185.1.

424.

In n. 385, 386, 387. egimus de viribus, juxta rationem, a ratione dupli-
catâ inverſa diſtantiæ parum aberrante, decreſcentibus, aut de curvis circulis fini-
timis; quia in hiſce caſibus in propoſitionibus error ſenſibilis non datur, licet vires
ſequantur rationem aliûs poteſtatis cujuſdam diſtantiæ; in quo caſu Mathe-
maticè loquendo curva non eſt Ellipſis mota juxta leges explicatas, ad quod
requiritur vis, quæ eſt ſumma aut differentia virium, quarum una ſequitur ra-
tionem inverſam duplicatam , alia inverſam triplicatam, diſtantiæ .

185.1.

425.
331. 411.
420.

Ut autem ex dato motu angulari Ellipſeos vim addendam aut detrahen-
dam, & vice verſa ex data hac, motum curvæ determinemus,
ſit A extremitas axeos majoris; F focus centrum virium; a A portio circuli
centro F, radio F A deſcripti; A L Ellipſeos portio.

185.1.

426.
TAB. XV.
fig, 13.

Ponamus dum corpus in Ellipſi fertur per AL, ipſam curvam motu an-
gulari a FA transferri; anguloſque a FL, AFL eſſe inter ſe ut M ad N. Ponimus etiam angulos hos eſſe infinite exiguos.

In a & A ad circulum a A ducantur tangentes a i, EAI, ſibi mutuo oc-
currentes in E, & quarum ultima etiam Ellipſin tangit in A; ducantur et-
iam AB, LI, ad a F parallelæ, ultima propter infinite exiguos arcus a A,
AL, pro parallela haberi poteſt ipſi AF; tandem ſint AC ad a B, & LG ad
AI parallelæ.

Sunt æquales E a, EA , ideoque a E & EB, quæ EA æqualis eſt. Pro- pter triangula ſimilia EBA, EiI, eſt #
# EB aut {1/2} a B, E i aut a i-{1/2} a B: : BA, i I; a B autem ſe habet ad a i, ut angulus a F A ad a FL, id eſt, ut M-N ad M: ergo
BA, i I: :{1/2} M-{1/2}N, {1/2}M + {1/2}N: :M-N, M + N.

185.1.

36. El. III.

Ex circuli proprietate a C aut BA, a A aut a B, & diameter, ſunt
in continua proportione ; ergo BA = {a B q /2AF}. Ellipſis in extremitate a- xeos majoris coincidit cum circulo cujus diameter eſt axeos parameter ; id-

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