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

156. SCHOLIUM. 3.
In quo quædam in boc capite memoratæ Cycloidis
proprietates demonſtrantur.

Poſitâ cycloidis memoratâ formatione; ſit circulus generator BEF. Pona- mus hunc perveniſſe ad punctum Gbaſeos, punctum F erit in f, poſito arcu Gf
lineæ GF æquali; Punctum deſcribens erit in b, & erit hoc punctum Cycloïdis-

156.1.

34.
282.
TAB. XII.
fig. 4.

Ducatur G c H diameter per punctum contactus, erit hæc ad baſin per-
pendicularis , & parallela diametro BF. Ductâ nunc b L, per punctum Cycloïdis b, baſi parallelâ, ſecante circulum FEB in E, & GH in I; ma-
nifeſtum eſt, propter æquales GI & FL , in circulis æqualibus æquales eſſe b I, EL; additâ utrimque IE æquales erunt b E, IL, cui æqualis
GF .

156.1.

18. El. III.
34. El. I.
34. El. I.

Facile etiam liquet arcus G f, b H, EB, æquales eſſe inter ſe & lineæ
GF; ideoque lineæ b E.

Ex quibus hanc curvæ deducimus proprietatem, Si ex puncto quocunque Cy-
cloidis ad baſin ducatur parallela, quæ ſemicirculum ſecat ſuper axe deſcriptum
ad partem curvæ, qualis linea hìc eſt b EL, erit hujus portio, inter Cycloi-
dem & ſemicirculum intercepta, æqualis arcui ſemicir culi inter lineam memora-
tam & verticem intercepto. id eſt b E arcui EB æqualis eſt.

156.1.

315.

Sit Cycloïs ADB; vertex B; baſis AF; axis BF, qui diameter eſt ſemi-
circuli FMB.

156.1.

316.
TAB. XII.
fig. 5.

Sumtâ D d portione quacunque infinitè exigua Cycloïdis, poterit hæc
pro lineâ rectâ haberi, & continuatâ formabit tangentem in puncto D aut d. Ducantur DL, dl, ad baſin parallelæ ſemicirculum ſecantes in E, e; & ductâ B e continuetur hæc donec ſecet in b lineam DL; ſit etiam BO ad ba-
ſin parallela, circulum tangens in B, & quæ in O ſecatur lineâ eO, con-
tinuatione lineæ E e.

Triangula b Ee & e OB, propter Bo & hE parallelas ſunt ſimilia. La-
tera autem EO & OB ſunt æqualia ; ergo & æqualia e E, h E; eſt eE ar- cuum B e BE, aut linearum de, DE, differentia ; quæ eadem differen- tia eſt ideò etiam h E, quare ſunt æquales parallelæ D h, de; ſuntetiam id-
circo æquales & parallelæ D d, b e . id eſt tangens in d parallela chordæ e B, quam Cycloïdis proprietatem ſuperius indicavimus in n. 285.

156.1.

36. El III.
315
33. El I.

Iiſdem poſitis ducatur FE i; erit hæc ad BE aut B b (propter augulum
infinite exiguum e BE) perpendicularis , dividetque baſin trianguli iſoce- les b E e in duas partes æquales ita, ut ei ſit dimidium ipſius eb aut d D. Eſt verò ei differentia inter chordas BE, Be; nam ſi centro B, radio BE,
circulus deſcribatur coincidet hic cum Ei, quæ infinite exigua eſt; & D d
eſt differentia arcuum Cycloidis DB, dB.

156.1.

317.
31. El. III,

Concipiamus nunc lineam ad baſim Cycloidis AF parallelam moveri à
B ad F, aliamque lineam interea circa B ita rotari, ut continuo tranſeat
per interſectionem primæ cum ſemicirculo. Ubi prima Ex. gr. pervenit
ad dl erit ſecunda in B e, translatâ primâ ad DL rotatur ſecunda ut ſit in
BE. In hoc motu, commune initium habent, & continuo augentur, arcus
Cycloïdis DB & chorda EB; ſed illius augmentum ſemper duplum eſt au-
gmentihujus, quare & integer arcus qui eſt ſumma augmentorum, erit du-
plusintegræ chordæ, quæ etiam ſummam valet augmentorum ſuorum. Ha-

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