1And becauſe in the Quadrilateral Figure I L B H the Sides H B and
H I are equal, and the Angles B and I Right Angles, the Side B L ſhall
likewiſe be equal to the Side L I: But E I is equal to E F: Therefore the
whole Line L E, or N E is equal to the two Lines L B and E F: Let
the Common Line E F be taken away, and the remainder F N ſhall be
equal to L B: And F B was ſuppoſed equal to B A: Therefore L B ſhall
be equal to the two Lines A B and B N. Again, if we ſuppoſe the
Time along A B to be the ſaid A B, the Time along E B ſhall be equal to
E B; and the Time along the whole E M ſhall be E N, namely, the
Mean-proportional betwixt M E and E B: I berefore the Time of the
Deſcent of the remaining part B M after E B, or after A B, ſhall be the
ſaid B N: But it hath been ſuppoſed, that the Time along A B is A B:
Therefore the Time of the Fall along both A B and B M is A B N:
And becauſe the Time along E B ex quiete in E is E B, the Time along
B M ex quiete in B ſhall be the Mean-proportional between B E and
B M; and this is B L: The Time, therefore, along both A B M ex quiete
in A is A B N: And the Time along B M only ex quiete in B is B L:
But it was proved that B L is equal to the two A B and B N: Therefore
the Propoſition is manifeſt.
H I are equal, and the Angles B and I Right Angles, the Side B L ſhall
likewiſe be equal to the Side L I: But E I is equal to E F: Therefore the
whole Line L E, or N E is equal to the two Lines L B and E F: Let
the Common Line E F be taken away, and the remainder F N ſhall be
equal to L B: And F B was ſuppoſed equal to B A: Therefore L B ſhall
be equal to the two Lines A B and B N. Again, if we ſuppoſe the
Time along A B to be the ſaid A B, the Time along E B ſhall be equal to
E B; and the Time along the whole E M ſhall be E N, namely, the
Mean-proportional betwixt M E and E B: I berefore the Time of the
Deſcent of the remaining part B M after E B, or after A B, ſhall be the
ſaid B N: But it hath been ſuppoſed, that the Time along A B is A B:
Therefore the Time of the Fall along both A B and B M is A B N:
And becauſe the Time along E B ex quiete in E is E B, the Time along
B M ex quiete in B ſhall be the Mean-proportional between B E and
B M; and this is B L: The Time, therefore, along both A B M ex quiete
in A is A B N: And the Time along B M only ex quiete in B is B L:
But it was proved that B L is equal to the two A B and B N: Therefore
the Propoſition is manifeſt.
Otherwiſe with more expedition.
Let B C be the Inclined Plane, and B A the Perpendicular. Continue
out C B to E, and unto E C erect a Perpendicular at B, which being
prolonged ſuppoſe B H equal to the exceſſe of B E above B A; and to the
Angle B H E let the Angle H E L be equal; and let E L continued out
meet with B K in L; and from L erect the Perpendicular L M unto E L
meeting B C in M. I ſay, that
135[Figure 135]
B M is the Space acquired in
the Plane B C. For becauſe
the Angle M L E is a Right
Angle, therefore B L ſhall be
a Mean-proportional betwixt
M B and B E; and L E a
Mean proportional betwixt M
E and E B; to which E L let
E N be cut equal: And the
three Lines N E, E L, and
L H ſhall be equal; and H B ſhall be the exceſſe of N E above B L: But
the ſaid H B is alſo the exceſſe of N E above N B and B A: Therefore
the two Lines N B and B A are equal to B L. And if we ſuppoſe E B
to be the Time along E B, B L ſhall be the Time along B M ex quiete in
B; and B N ſhall be the Time of the ſame B M after E B or after A B;
and A B ſhall be the Time along A B: Therefore the Times along A B M,
namely, A B N, are equal to the Times along the ſole Line B M ex quiete
in B: Which was intended.
out C B to E, and unto E C erect a Perpendicular at B, which being
prolonged ſuppoſe B H equal to the exceſſe of B E above B A; and to the
Angle B H E let the Angle H E L be equal; and let E L continued out
meet with B K in L; and from L erect the Perpendicular L M unto E L
meeting B C in M. I ſay, that
135[Figure 135]B M is the Space acquired in
the Plane B C. For becauſe
the Angle M L E is a Right
Angle, therefore B L ſhall be
a Mean-proportional betwixt
M B and B E; and L E a
Mean proportional betwixt M
E and E B; to which E L let
E N be cut equal: And the
three Lines N E, E L, and
L H ſhall be equal; and H B ſhall be the exceſſe of N E above B L: But
the ſaid H B is alſo the exceſſe of N E above N B and B A: Therefore
the two Lines N B and B A are equal to B L. And if we ſuppoſe E B
to be the Time along E B, B L ſhall be the Time along B M ex quiete in
B; and B N ſhall be the Time of the ſame B M after E B or after A B;
and A B ſhall be the Time along A B: Therefore the Times along A B M,
namely, A B N, are equal to the Times along the ſole Line B M ex quiete
in B: Which was intended.
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