PROBL. III. PROP. VI.
The Sublimity and Altitude of a Semiparabola
being given to find its Amplitude.
being given to find its Amplitude.
Let A C be perpendicular to the
155[Figure 155]
Horizontal Line D C, in
which let the Altitude C B and
the Sublimity B A be given: It is
required in the Horizontal Line
D C to find the Amplitude of the
Semiparabola that is deſcribed out of
the Sublimity B A with the Alti
tude B C. Take a Mean proportional
between C B and B A, to which let
C D be double, I ſay, that C D is
the Amplitude required. The which
is manifeſt by the precedent Propoſition.
155[Figure 155]Horizontal Line D C, in
which let the Altitude C B and
the Sublimity B A be given: It is
required in the Horizontal Line
D C to find the Amplitude of the
Semiparabola that is deſcribed out of
the Sublimity B A with the Alti
tude B C. Take a Mean proportional
between C B and B A, to which let
C D be double, I ſay, that C D is
the Amplitude required. The which
is manifeſt by the precedent Propoſition.
THEOR. IV. PROP. VII.
In Projects which deſcribe Semiparabola's of the
ſame Amplitude, there is leſs Impetus required
in that which deſcribeth that whoſe Ampli
tude is double to its Altitude, than in any
other.
ſame Amplitude, there is leſs Impetus required
in that which deſcribeth that whoſe Ampli
tude is double to its Altitude, than in any
other.
For let the Semiparabola be B D, whoſe Amplitude C D is dou
ble to its Altitude C B; and in its Axis extended on high let B A
be ſuppoſed equal to the Altitude B C; and draw a Line from
A to D which toucheth the Semiparabola in D, and ſhall cut the Hori
zontal Line B E in E; and B E ſhall be equal to B C or to B A: It is
manifeſt that it is deſcribed by the Project whoſe Equable Horizontal
Impetus is ſuch as is that gained in B of a thing falling from Reſt in A,
and the Impetus of the Natural Motion downwards, ſuch as is that of
a thing coming to C ex quiete in B. Whence it is manifeſt, that the
Impetus compounded of them, and that ſtriketh in the Term D is as the
Diagonal A E, that is potentia equal to them both. Now let there be
another Semiparabola G D, whoſe Amplitude is the ſame C D, and the
Altitude C G leſs, or greater than the Altitude B C, and let H D touch
the ſame, cutting the Horizontal Line drawn by G in the point K; and
as H G is to G K, ſo let K G be to G L: by what hath been demonſtrated
G L ſhall be the Altitude from which the Project falling deſcribeth the
ble to its Altitude C B; and in its Axis extended on high let B A
be ſuppoſed equal to the Altitude B C; and draw a Line from
A to D which toucheth the Semiparabola in D, and ſhall cut the Hori
zontal Line B E in E; and B E ſhall be equal to B C or to B A: It is
manifeſt that it is deſcribed by the Project whoſe Equable Horizontal
Impetus is ſuch as is that gained in B of a thing falling from Reſt in A,
and the Impetus of the Natural Motion downwards, ſuch as is that of
a thing coming to C ex quiete in B. Whence it is manifeſt, that the
Impetus compounded of them, and that ſtriketh in the Term D is as the
Diagonal A E, that is potentia equal to them both. Now let there be
another Semiparabola G D, whoſe Amplitude is the ſame C D, and the
Altitude C G leſs, or greater than the Altitude B C, and let H D touch
the ſame, cutting the Horizontal Line drawn by G in the point K; and
as H G is to G K, ſo let K G be to G L: by what hath been demonſtrated
G L ſhall be the Altitude from which the Project falling deſcribeth the