289187MATHEMATICA. LIB. I. CAP. XXVIII.
de colliſione agatur, corpora tantum concurrentia conſide-
ramus.
ramus.
SCHOLIUM 1.
Demonſtratio n. 660.
QUamdiu corpora moventur in eâdem lineâ propoſitio ultimum memo-
11662. rata ſimplici algebraica computatione patet.
11662. rata ſimplici algebraica computatione patet.
Sint corpora A, B, C, primi velocitas m;
ſecundi n, tertii p;
centri gravi-
tatis velocitas d. Tendant corpora ad eandem partem; & ſint m & n majores
ipſa d; p verò minor: Ergo velocitates, quibus corpora ad centrum gravitatis
tendunt ſunt m - d, n - d, d - p; & A x m - d + B x n - d = C x d - p; 22654. 2 A md - 2A dd + 2B nd - 2 B dd = 2 C dd - 2C dp, multiplicando inte-
gram æquationem per 2d. Demonſtrandum A mm + B nn + C pp = A + B + C
x dd + A x m - d2 + B x n - d2 + C x d - p2. Ultima hæc quantitas ſic pot-
eſt exprimi A mm-2 A md + 2 A dd + B nn - 2B nd + 2 B dd + C pp
- 2 C pd + 2C dd. Sed - 2A md + 2A dd - 2B nd + 2B dd & - 2C pd
+ 2 C dd ſeſe mutuo deſtruunt & quantitas hæc tantum valet A mm + B nn
+ C pp. Quod demonſtrandum erat.
tatis velocitas d. Tendant corpora ad eandem partem; & ſint m & n majores
ipſa d; p verò minor: Ergo velocitates, quibus corpora ad centrum gravitatis
tendunt ſunt m - d, n - d, d - p; & A x m - d + B x n - d = C x d - p; 22654. 2 A md - 2A dd + 2B nd - 2 B dd = 2 C dd - 2C dp, multiplicando inte-
gram æquationem per 2d. Demonſtrandum A mm + B nn + C pp = A + B + C
x dd + A x m - d2 + B x n - d2 + C x d - p2. Ultima hæc quantitas ſic pot-
eſt exprimi A mm-2 A md + 2 A dd + B nn - 2B nd + 2 B dd + C pp
- 2 C pd + 2C dd. Sed - 2A md + 2A dd - 2B nd + 2B dd & - 2C pd
+ 2 C dd ſeſe mutuo deſtruunt & quantitas hæc tantum valet A mm + B nn
+ C pp. Quod demonſtrandum erat.
Sint iterum tria corpora A, B, C, quorum tantum gravitatis centra conſi-
33663. deramus; ſit commune gravitatis centrum D; ponamus corpora moveri per
44TA. XXV.
fig. 10. AE, BE, CF, velocitatibus hiſce lineis proportionalibus. Directio & ce-
leritas centri gravitatis D eſt DE. Velocitates, quibus corpora ad centrum
commune gravitatis tendunt, ſunt AD, BD, CD, hæ enim eſſent corpo-
rum velocitates in nave, in qua centrum gravitatis quieſceret. Idcirco de-
monſtrandum A x AEq + B x BEq + C x CEq = A + B + C x DEq + A x ADq
+ B x BDq + C x CDq.
33663. deramus; ſit commune gravitatis centrum D; ponamus corpora moveri per
44TA. XXV.
fig. 10. AE, BE, CF, velocitatibus hiſce lineis proportionalibus. Directio & ce-
leritas centri gravitatis D eſt DE. Velocitates, quibus corpora ad centrum
commune gravitatis tendunt, ſunt AD, BD, CD, hæ enim eſſent corpo-
rum velocitates in nave, in qua centrum gravitatis quieſceret. Idcirco de-
monſtrandum A x AEq + B x BEq + C x CEq = A + B + C x DEq + A x ADq
+ B x BDq + C x CDq.
Ad DE ducantur perpendieulares AF, BG, CH, LDL.
Diſtantiæ
corporum A, B, C à linea LDL ſunt FD, GD, HD; ergo, quia D eſt
centrum commune gravitatis A x FD + B x GD = C x D unde patet 55141. 159 eorum corporum eſſe commune gravitatis centrum poſitis his in F, G
& H . Si in hoc ſitu concipiamus corpora moveri A velocitate FE, 66141. velocitate GE, & tandem C velocitate HE; centri gravitatis velocitas
erit DE; Ergo A x FEq + B x GEq + C x HEq = A + B + C x DEq
+ A x FDq + B x GDq + C x HDq addendo utrimque A x AFq 77661. B x BGq + C x CHq & ſubſtituendo triangulorum rectangulorum AFD,
BGD, CHD, AFE, BGE, CHE, quadrata Hypotenuſarum pro
quadratis laterum , habebimus propoſitum. 8847. EL @
corporum A, B, C à linea LDL ſunt FD, GD, HD; ergo, quia D eſt
centrum commune gravitatis A x FD + B x GD = C x D unde patet 55141. 159 eorum corporum eſſe commune gravitatis centrum poſitis his in F, G
& H . Si in hoc ſitu concipiamus corpora moveri A velocitate FE, 66141. velocitate GE, & tandem C velocitate HE; centri gravitatis velocitas
erit DE; Ergo A x FEq + B x GEq + C x HEq = A + B + C x DEq
+ A x FDq + B x GDq + C x HDq addendo utrimque A x AFq 77661. B x BGq + C x CHq & ſubſtituendo triangulorum rectangulorum AFD,
BGD, CHD, AFE, BGE, CHE, quadrata Hypotenuſarum pro
quadratis laterum , habebimus propoſitum. 8847. EL @
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