PROBL. IV. PROP. XI.
The Impetus and Amplitude of a Semiparabola be
ing given, to find its Altitude, and conſequently
its Sublimity.
ing given, to find its Altitude, and conſequently
its Sublimity.
Let the Impetus given be defined by the Perpendicular to the Ho
rizon A B; and let the Amplitude along the Horizontal Line be
B C. It is required to find the Altitude and Sublimity of the
Parabola whoſe Impetus is A B, and Amplitude B C. It is manifeſt,
from what hath been already demonſtrated, that half the Amplitude B C
will be a Mean-proportional betwixt the Altitude and the Sublimity of
the ſaid Semiparabola, whoſe Impetus, by the precedent Propoſition, is
the ſame with the Impetus of the Moveable falling from Reſt in A along
the whole Perpendicular A B: Wherefore B A is ſo to be cut that the
Rectangle contained by its parts may be equal to the Square of half of
B C, which let be B D. Hence it appeareth
to be neceſſary that D B do not exceed the
161[Figure 161]
half of B A; for of Rectangles contained by
the parts the greateſt is when the whole
Line is cut into two equal parts. Therefore
let B A be divided into two equal parts in E.
And if B D be equal to B E the work is
done; and the Altitude of the Semipara
bola ſhall be B E, and its Sublimity E A:
(and ſee here by the way that the Amplitude
of the Parabola of a Semi-right Elevation,
as was demonſtrated above, is the greateſt of
all thoſe deſcribed with the ſame Impetus.)
But let B D be leſs than the half of B A,
which is ſo to be cut that the Rectangle under the parts may be equal to
the Square B D. Upon E A deſcribe a Semicircle, upon which out of A
ſet off A F equal to B D, and draw a Line from F to E, to which cut
a part equal E G. Now the Rectangle B G A, together with the Square
E G, ſhall be equal to the Square E A; to which the two Squares A F
and F E are alſo equal: Therefore the equal Squares G E and F E be
ing ſubſtracted, there remaineth the Rectangle B G A equal to the
Square A F, ſcilicet, to B D; and the Line B D is a Mean-proportional
betwixt B G and G A. Whence it appeareth, that of the Semipa
rabola whoſe Amplitude is B C, and Impetus A B, the Altitude is
B G, and the Sublimity G A. And if we ſet off B I below equal to G A,
this ſhall be the Altitude, and I A the Sublimity of the Semiparabola
I C. From what hath been already demonſtrated we are able,
rizon A B; and let the Amplitude along the Horizontal Line be
B C. It is required to find the Altitude and Sublimity of the
Parabola whoſe Impetus is A B, and Amplitude B C. It is manifeſt,
from what hath been already demonſtrated, that half the Amplitude B C
will be a Mean-proportional betwixt the Altitude and the Sublimity of
the ſaid Semiparabola, whoſe Impetus, by the precedent Propoſition, is
the ſame with the Impetus of the Moveable falling from Reſt in A along
the whole Perpendicular A B: Wherefore B A is ſo to be cut that the
Rectangle contained by its parts may be equal to the Square of half of
B C, which let be B D. Hence it appeareth
to be neceſſary that D B do not exceed the
161[Figure 161]half of B A; for of Rectangles contained by
the parts the greateſt is when the whole
Line is cut into two equal parts. Therefore
let B A be divided into two equal parts in E.
And if B D be equal to B E the work is
done; and the Altitude of the Semipara
bola ſhall be B E, and its Sublimity E A:
(and ſee here by the way that the Amplitude
of the Parabola of a Semi-right Elevation,
as was demonſtrated above, is the greateſt of
all thoſe deſcribed with the ſame Impetus.)
But let B D be leſs than the half of B A,
which is ſo to be cut that the Rectangle under the parts may be equal to
the Square B D. Upon E A deſcribe a Semicircle, upon which out of A
ſet off A F equal to B D, and draw a Line from F to E, to which cut
a part equal E G. Now the Rectangle B G A, together with the Square
E G, ſhall be equal to the Square E A; to which the two Squares A F
and F E are alſo equal: Therefore the equal Squares G E and F E be
ing ſubſtracted, there remaineth the Rectangle B G A equal to the
Square A F, ſcilicet, to B D; and the Line B D is a Mean-proportional
betwixt B G and G A. Whence it appeareth, that of the Semipa
rabola whoſe Amplitude is B C, and Impetus A B, the Altitude is
B G, and the Sublimity G A. And if we ſet off B I below equal to G A,
this ſhall be the Altitude, and I A the Sublimity of the Semiparabola
I C. From what hath been already demonſtrated we are able,