THEOR. VII. PROP. X.
The Impetus or Moment of any Semiparabola is
equal to the Moment of any Moveable falling
naturally along the Perpendicular to the Ho
rizon that is equal to the Line compounded of
the Sublimity and of the Altitude of the Se
miparabola.
equal to the Moment of any Moveable falling
naturally along the Perpendicular to the Ho
rizon that is equal to the Line compounded of
the Sublimity and of the Altitude of the Se
miparabola.
Let the Semiparabola be A B, its Sublimity D A, and Altitude
A C, of which the Perpendicular D C is compounded. I ſay, that
the Impetus of the Semiparabola in B is equal to the Moment of
the Moveable Naturally falling from D to C. Suppoſe D C it ſelf to be
the Meaſure of the Time and of the Impetus; and take a Mean-pro
portional betwixt C D and D A, to which let
160[Figure 160]
C F be equal; and withal let C E be a Mean
proportional between D C and C A: Now C F
ſhall be the Meaſure of the Time and of the Mo
ment of the Moveable ſalling along D A out of
Reſt in D; and C E ſhall be the Time and Mo
ment of the Moveable falling along A C, out of
Reſt in A, and the Moment of the Diagonal E F
ſhall be that compounded of both the others, ſcil.
that of the Semiparabola in B. And becauſe
D C is cut according to any proportion in A, and becauſe C F and C E
are Mean-Proportionals between C D and the parts D A and A C; the
Squares of them taken together ſhall be equal to the Square of the
whole; by the Lemma aforegoing: But the Squares of them are alſo
equal to the Square of E F: Therefore D F is equal alſo to the Line D C:
Whence it is manifeſt that the Moments along D C, and along the Se
miparabola A B, are equal in C and B: Which was required.
A C, of which the Perpendicular D C is compounded. I ſay, that
the Impetus of the Semiparabola in B is equal to the Moment of
the Moveable Naturally falling from D to C. Suppoſe D C it ſelf to be
the Meaſure of the Time and of the Impetus; and take a Mean-pro
portional betwixt C D and D A, to which let
160[Figure 160]C F be equal; and withal let C E be a Mean
proportional between D C and C A: Now C F
ſhall be the Meaſure of the Time and of the Mo
ment of the Moveable ſalling along D A out of
Reſt in D; and C E ſhall be the Time and Mo
ment of the Moveable falling along A C, out of
Reſt in A, and the Moment of the Diagonal E F
ſhall be that compounded of both the others, ſcil.
that of the Semiparabola in B. And becauſe
D C is cut according to any proportion in A, and becauſe C F and C E
are Mean-Proportionals between C D and the parts D A and A C; the
Squares of them taken together ſhall be equal to the Square of the
whole; by the Lemma aforegoing: But the Squares of them are alſo
equal to the Square of E F: Therefore D F is equal alſo to the Line D C:
Whence it is manifeſt that the Moments along D C, and along the Se
miparabola A B, are equal in C and B: Which was required.
COROLLARY.
Hence it is manifeſt, that of all Parabola's whoſe Altitudes and
Sublimities being joyned together are equal, the Impetus's are
alſo equal.
Sublimities being joyned together are equal, the Impetus's are
alſo equal.
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