1Second Book. You muſt alſo know, that the point F which divi
deth the Tangent E B in the middle, will many other times fall
above the point A, and once alſo in the ſaid A: In which caſes it is
evident of it ſelf, that the third proportional to the half of the Tan
gent, and to B I (which giveth the Sublimity) is all above A. But
the Author hath taken a Caſe in which it was not manifeſt that the
ſaid third Proportional is alwaies greater than F A: and which
therefore being ſet off above the point F paſſeth beyond the Paral
lel A G. Now let us proceed.
deth the Tangent E B in the middle, will many other times fall
above the point A, and once alſo in the ſaid A: In which caſes it is
evident of it ſelf, that the third proportional to the half of the Tan
gent, and to B I (which giveth the Sublimity) is all above A. But
the Author hath taken a Caſe in which it was not manifeſt that the
ſaid third Proportional is alwaies greater than F A: and which
therefore being ſet off above the point F paſſeth beyond the Paral
lel A G. Now let us proceed.
It will not be unprofitable if by help of this Table we compoſe ano
ther, ſhewing the Altitudes of the ſame Semiparabola's of Projects of
the ſame Impetus. And the Conſtruction of it is in this manner.
ther, ſhewing the Altitudes of the ſame Semiparabola's of Projects of
the ſame Impetus. And the Conſtruction of it is in this manner.
PROBL. VI. PROP. XIII.
From the given Amplitudes of Semiparabola's in
the following Table ſet down, keeping the
common Impeius with which every one of
them is deſcribed, to compute the Altitudes of
each ſeveral Semiparabola.
the following Table ſet down, keeping the
common Impeius with which every one of
them is deſcribed, to compute the Altitudes of
each ſeveral Semiparabola.
Let the Amplitude given be B C, and of the Impetus, which is
ſuppoſed to be alwaies the ſame, let the Meaſure be O B, to wit,
the Aggregate of the Altitude and Sublimity. The ſaid Altitude
is required to be found and diſtinguiſhed. Which ſhall then be done when
B O is ſo divided as that the Rectangle contained under its parts is
equal to the Square of half the Amplitude B C. Let that ſame divi
ſion fall in F; and let both O B and B C be cut in the midſt at D and I.
163[Figure 163]
The Square I B, therefore, is equal to the
Rectangle B F O: And the Square D O is
equal to the ſame Rectangle together with the
Square F D. If therefore from the Square
D O we deduct the Square B I, which is equal
to the Rectangle B F O, there ſhall remain
the Square F D; to whoſe Side D F, B D be
ing added it ſhall give the deſired Altitude
Altitude B F. And it is thus compounded
ex datis. From half of the Square B O known
ſubſtract the Square B I alſo known, of the remainder take the Square
Root, to which add D B known; and you ſhall have the Altitude ſought
B F. For example. The Altitude of the Parabola deſcribed at the
Elevation of 55 degrees is to be found. The Amplitude, by the follow
ing Table is 9396, its half is 4698, the Square of that is 22071204,
ſuppoſed to be alwaies the ſame, let the Meaſure be O B, to wit,
the Aggregate of the Altitude and Sublimity. The ſaid Altitude
is required to be found and diſtinguiſhed. Which ſhall then be done when
B O is ſo divided as that the Rectangle contained under its parts is
equal to the Square of half the Amplitude B C. Let that ſame divi
ſion fall in F; and let both O B and B C be cut in the midſt at D and I.
163[Figure 163]The Square I B, therefore, is equal to the
Rectangle B F O: And the Square D O is
equal to the ſame Rectangle together with the
Square F D. If therefore from the Square
D O we deduct the Square B I, which is equal
to the Rectangle B F O, there ſhall remain
the Square F D; to whoſe Side D F, B D be
ing added it ſhall give the deſired Altitude
Altitude B F. And it is thus compounded
ex datis. From half of the Square B O known
ſubſtract the Square B I alſo known, of the remainder take the Square
Root, to which add D B known; and you ſhall have the Altitude ſought
B F. For example. The Altitude of the Parabola deſcribed at the
Elevation of 55 degrees is to be found. The Amplitude, by the follow
ing Table is 9396, its half is 4698, the Square of that is 22071204,
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