PROBL. V. PROP. XII.
To collect by Calculation of the Amplitudes of all
Semiparabola's that are deſcribed by Projects
expulſed with the ſame Impetus, and to make
Tables thereof.
Semiparabola's that are deſcribed by Projects
expulſed with the ſame Impetus, and to make
Tables thereof.
It is obvious, from the things demonſtrated, that Parabola's are de
ſcribed by Projects of the ſame Impetus then, when their Subli
mities together with their Altitudes do make up equal Perpendicu
lars upon the Horizon. Theſe Perpendiculars therefore are to be com
prehended between the ſame Horizontal Parallels. Therefore let the
Horizontal Line C B be ſuppoſed equal to the Perpendicular B A, and
draw the Diagonal from A to C. The Angle A C B ſhall be Semi
right, or 45 Degrees. And the Perpendicular B A being divided into
two equal parts in D, the Semiparabola D C ſhall be that which is de
ſcribed from the Sublimity A D together with the Altitude D B: and
its Impetus in C ſhall be as great as that of the Moveable coming out of
Reſt in A along the Perpendicular A B is in B. And if A G be drawn
parallel to B C, the united Altitudes and Sublimities of all other re
maining Semiparabola's whoſe future Impetus's are the ſame with thoſe
now mentioned muſt be bounded by the Space between the Parallels
162[Figure 162]
A G and B C. Farthermore, it having
been but now demonſtrated, that the Am
plitudes of the Semiparabola's whoſe
Tangents are equidiſtant either above or
below from the Semi right Elevation are
equal, the Calculations that we frame
for the greater Elevations will likewiſe
ſerve for the leſſer. We chooſe moreover
a number of ten thouſand parts for the
greateſt Amplitude of the Projection of
the Semiparabola made at the Elevation
of 45 degrees: ſo much therefore the Line
B A, and the Amplitude of the Semipa
rabola B C, are to be ſuppoſed. And we
make choice of the number 10000, becauſe we in our Calculation uſe
the Table of Tangents, in which this number agreeth with the Tangent
of 45 degrees. Now, to come to the buſineſs, let C E be drawn, contain
ing the Angle E C B greater (Acute nevertheleſs,) than the Angle
A C B; and let the Semiparabola be deſcribed which is touched by the
Line E C, and whoſe Sublimity united with its Altitude is equal to
B A. In the Table of Tangents take the ſaid B E for the Tangent at the
ſcribed by Projects of the ſame Impetus then, when their Subli
mities together with their Altitudes do make up equal Perpendicu
lars upon the Horizon. Theſe Perpendiculars therefore are to be com
prehended between the ſame Horizontal Parallels. Therefore let the
Horizontal Line C B be ſuppoſed equal to the Perpendicular B A, and
draw the Diagonal from A to C. The Angle A C B ſhall be Semi
right, or 45 Degrees. And the Perpendicular B A being divided into
two equal parts in D, the Semiparabola D C ſhall be that which is de
ſcribed from the Sublimity A D together with the Altitude D B: and
its Impetus in C ſhall be as great as that of the Moveable coming out of
Reſt in A along the Perpendicular A B is in B. And if A G be drawn
parallel to B C, the united Altitudes and Sublimities of all other re
maining Semiparabola's whoſe future Impetus's are the ſame with thoſe
now mentioned muſt be bounded by the Space between the Parallels
162[Figure 162]A G and B C. Farthermore, it having
been but now demonſtrated, that the Am
plitudes of the Semiparabola's whoſe
Tangents are equidiſtant either above or
below from the Semi right Elevation are
equal, the Calculations that we frame
for the greater Elevations will likewiſe
ſerve for the leſſer. We chooſe moreover
a number of ten thouſand parts for the
greateſt Amplitude of the Projection of
the Semiparabola made at the Elevation
of 45 degrees: ſo much therefore the Line
B A, and the Amplitude of the Semipa
rabola B C, are to be ſuppoſed. And we
make choice of the number 10000, becauſe we in our Calculation uſe
the Table of Tangents, in which this number agreeth with the Tangent
of 45 degrees. Now, to come to the buſineſs, let C E be drawn, contain
ing the Angle E C B greater (Acute nevertheleſs,) than the Angle
A C B; and let the Semiparabola be deſcribed which is touched by the
Line E C, and whoſe Sublimity united with its Altitude is equal to
B A. In the Table of Tangents take the ſaid B E for the Tangent at the
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