* Or Angle of
45.
45.
Of the Triangle M C B, about the Right-Angle C, let the Ho
rizontal Line B C and the Perpendicular C M be equal; for
ſo the Angle M B C ſhall be Semi-right; and prolonging C M
to D, let there be conſtituted in B two equal Angles above and below
the Diagonal M B, viz. M B E, and M B D. It is to be demonſtrated
that the Amplitudes of the Parabola's deſcribed by the Projects be
ing emitted [or ſhot off] with the ſame Impetus out of the Term B,
according to the Elevations of the Angles E B C and D B C, are equal.
For in regard that the extern Angle B M C, is equal to the two intern
M D B and M B D, the Angle M B C ſhall alſo be equal to them. And if
we ſuppoſe M B E inſtead of the Angle M B D,
the ſaid Angle M B C ſhall be equal to the two
157[Figure 157]
Angles M B E and B D C: And taking away
the common Angle M B E, the remaining An
gle B D C ſhall be equal to the remaining An
gle E B C: Therefore the Triangles D C B
and B C E are alike. Let the Right Lines
D C and E C be divided in the midſt in H and
F; and draw H I and F G parallel to the Ho
rizontal Line C B; and as D H is to H I, ſo
let I H be to H L: the Triangle I H L ſhall be
like to the Triangle I H D, like to which alſo is E G F. And ſeeing
that I H and G F are equal (to wit, halves of the ſame B C:) There
fore F E, that is F C, ſhall be equal to H L: And, adding the common
Line F H, C H ſhall be equal to F L. If therefore we underſtand the Se
miparabola to be deſcribed along by H and B, whoſe Altitude ſhall be
H C, and Sublimity H L, its Amplitude ſhall be C B, which is double
to HI, that is, the Mean betwixt D H, or C H, and HL: And D B
ſhall be a Tangent to it, the Lines C H and H D being equal. And if,
again, we conceive the Parabola to be deſcribed along by F and B from
the Sublimity FL, with the Altitude F C, betwixt which the Mean
proportional is F G, whoſe double is the Horizontal Line C B: C B, as
before, ſhall be its Amplitude; and E B a Tangent to it, ſince E F and
F C are equal: But the Angles D B C and E B C (ſcilicet, their Eleva
tions) ſhall be equidiſtant from the Semi-Right Angle: Therefore the
Propoſition is demonſtrated.
rizontal Line B C and the Perpendicular C M be equal; for
ſo the Angle M B C ſhall be Semi-right; and prolonging C M
to D, let there be conſtituted in B two equal Angles above and below
the Diagonal M B, viz. M B E, and M B D. It is to be demonſtrated
that the Amplitudes of the Parabola's deſcribed by the Projects be
ing emitted [or ſhot off] with the ſame Impetus out of the Term B,
according to the Elevations of the Angles E B C and D B C, are equal.
For in regard that the extern Angle B M C, is equal to the two intern
M D B and M B D, the Angle M B C ſhall alſo be equal to them. And if
we ſuppoſe M B E inſtead of the Angle M B D,
the ſaid Angle M B C ſhall be equal to the two
157[Figure 157]Angles M B E and B D C: And taking away
the common Angle M B E, the remaining An
gle B D C ſhall be equal to the remaining An
gle E B C: Therefore the Triangles D C B
and B C E are alike. Let the Right Lines
D C and E C be divided in the midſt in H and
F; and draw H I and F G parallel to the Ho
rizontal Line C B; and as D H is to H I, ſo
let I H be to H L: the Triangle I H L ſhall be
like to the Triangle I H D, like to which alſo is E G F. And ſeeing
that I H and G F are equal (to wit, halves of the ſame B C:) There
fore F E, that is F C, ſhall be equal to H L: And, adding the common
Line F H, C H ſhall be equal to F L. If therefore we underſtand the Se
miparabola to be deſcribed along by H and B, whoſe Altitude ſhall be
H C, and Sublimity H L, its Amplitude ſhall be C B, which is double
to HI, that is, the Mean betwixt D H, or C H, and HL: And D B
ſhall be a Tangent to it, the Lines C H and H D being equal. And if,
again, we conceive the Parabola to be deſcribed along by F and B from
the Sublimity FL, with the Altitude F C, betwixt which the Mean
proportional is F G, whoſe double is the Horizontal Line C B: C B, as
before, ſhall be its Amplitude; and E B a Tangent to it, ſince E F and
F C are equal: But the Angles D B C and E B C (ſcilicet, their Eleva
tions) ſhall be equidiſtant from the Semi-Right Angle: Therefore the
Propoſition is demonſtrated.
THEOR. VI. PROP. IX.
The Amplitudes of Parabola's, whoſe Altitudes
and Sublimities anſwer to each other è contra
rio, are equall.
and Sublimities anſwer to each other è contra
rio, are equall.
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