1Parabola G D. Let G M be a Mean-proportional betwixt A B and
G L; G M ſhall be the Time, and the Moment or Impetus in G of the
Project falling from L, (for it hath been ſuppoſed that A B is the Mea
ſure of the Time and Impetus.) Again, let G N be a Mean-propor
tional betwixt B C and C G: this G N ſhall be the Meaſure of the
Time and the
Impetus of the
156[Figure 156]
Project falling
from G to C.
If therefore a
Line be drawn
from M to N
it ſhall be the
the Meaſure of
the Impetus of
the Project a
long the Para
bola B D, ſcri
king in the
term D. Which
Impetus, I ſay,
is greater than the Impetus of the Project along the Parabola B D,
whoſe quantity was A E. For becauſe G N is ſuppoſed the Mean-pro
portional betwixt B C and C G, and B C is equal to B E, that is to H G;
(for they are each of them ſubduple to D C:) Therefore as C G is to
G N, ſo ſhall N G be to G K: and, as C G or H G is to G K, ſo ſhall the
Square N G be to the Square of G K: But as H G is to G K, ſo was
K G ſuppoſed to be to G L: Therefore as N G is to the Square G K, ſo
is K G to G L: But as K G is to G L, ſo is the Square K G unto the
Square G M, (for G M is the Mean between K G and G L:) Therefore
the three Squares N G, K G, and G M are continual proportionals: And
the two extream ones N G and G M taken together, that is the Square
M N is greater than double the Square K G, to which the Square A E
is double: Therefore the Square M N is greater than the Square A E:
and the Line M N greater than the Line A E: Which was to be de
monſtrated.
G L; G M ſhall be the Time, and the Moment or Impetus in G of the
Project falling from L, (for it hath been ſuppoſed that A B is the Mea
ſure of the Time and Impetus.) Again, let G N be a Mean-propor
tional betwixt B C and C G: this G N ſhall be the Meaſure of the
Time and the
Impetus of the
156[Figure 156]Project falling
from G to C.
If therefore a
Line be drawn
from M to N
it ſhall be the
the Meaſure of
the Impetus of
the Project a
long the Para
bola B D, ſcri
king in the
term D. Which
Impetus, I ſay,
is greater than the Impetus of the Project along the Parabola B D,
whoſe quantity was A E. For becauſe G N is ſuppoſed the Mean-pro
portional betwixt B C and C G, and B C is equal to B E, that is to H G;
(for they are each of them ſubduple to D C:) Therefore as C G is to
G N, ſo ſhall N G be to G K: and, as C G or H G is to G K, ſo ſhall the
Square N G be to the Square of G K: But as H G is to G K, ſo was
K G ſuppoſed to be to G L: Therefore as N G is to the Square G K, ſo
is K G to G L: But as K G is to G L, ſo is the Square K G unto the
Square G M, (for G M is the Mean between K G and G L:) Therefore
the three Squares N G, K G, and G M are continual proportionals: And
the two extream ones N G and G M taken together, that is the Square
M N is greater than double the Square K G, to which the Square A E
is double: Therefore the Square M N is greater than the Square A E:
and the Line M N greater than the Line A E: Which was to be de
monſtrated.
CORROLLARY I.
Hence it appeareth, that on the contrary, in the Project out of D
along the Semiparabola D B, leſs Impetus is required than
along any other according to the greater or leſſer Elevation
of the Semiparabola B D, which is according to the Tan
gent A D, containing half a Right-Angle upon the Hori
zon.
along the Semiparabola D B, leſs Impetus is required than
along any other according to the greater or leſſer Elevation
of the Semiparabola B D, which is according to the Tan
gent A D, containing half a Right-Angle upon the Hori
zon.
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