THEOR. IV. PROP. IV.
The Times of the Motions along equal Planes,
but unequally inclined, are to each other in
ſubduple proportion of the Elevations of thoſe
Planes Reciprocally taken.
but unequally inclined, are to each other in
ſubduple proportion of the Elevations of thoſe
Planes Reciprocally taken.
Let there proceed from the term B two equal Planes, but une
qually inclined, B A and B C, and let A E and C D be Hori
zontal Lines, drawn as far as the Perpendicular B D: Let the
Elevation of the Plane B A be B E; and let the Elevation of the
Plane B C be B D: And let B I be a Mean Proportional between the
Elevations D B and B E: It is manifeſt
91[Figure 91]
that the proportion of D B to B I, is ſub
duple the proportion of D B to B E. Now
I ſay, that the proportion of the Times
of the Deſcents or Motions along the
Planes B A and B C, are the ſame with
the proportion of D B to B I Reciprocal
ly taken: So that to the Time B A the
Elevation of the other Plane B C, that is
B D be Homologal; and to the Time along
B C, B I be Homologal: Therefore it is
to be demonſtrated, That the Time along B A is to the Time along
B C, as D B is to B I. Let I S be drawn equidiſtant from D C. And
becauſe it hath been demonſtrated that the Time of the Deſcent
along B A, is to the Time of the Deſcent along the Perpendicular
B E, as the ſaid B A is to B E; and the Time along B E is to the
Time along B D, as B E is to B I; and the Time along B D is to the
Time along B C, as B D to B C, or as B I to B S: Therefore, ex æqua
li, the Time along B A ſhall be to the Time along B C as B A to B S,
or as C B to BS: But C B is to B S, as D B to B I: Therefore the
Propoſition is manifeſt:
qually inclined, B A and B C, and let A E and C D be Hori
zontal Lines, drawn as far as the Perpendicular B D: Let the
Elevation of the Plane B A be B E; and let the Elevation of the
Plane B C be B D: And let B I be a Mean Proportional between the
Elevations D B and B E: It is manifeſt
91[Figure 91]that the proportion of D B to B I, is ſub
duple the proportion of D B to B E. Now
I ſay, that the proportion of the Times
of the Deſcents or Motions along the
Planes B A and B C, are the ſame with
the proportion of D B to B I Reciprocal
ly taken: So that to the Time B A the
Elevation of the other Plane B C, that is
B D be Homologal; and to the Time along
B C, B I be Homologal: Therefore it is
to be demonſtrated, That the Time along B A is to the Time along
B C, as D B is to B I. Let I S be drawn equidiſtant from D C. And
becauſe it hath been demonſtrated that the Time of the Deſcent
along B A, is to the Time of the Deſcent along the Perpendicular
B E, as the ſaid B A is to B E; and the Time along B E is to the
Time along B D, as B E is to B I; and the Time along B D is to the
Time along B C, as B D to B C, or as B I to B S: Therefore, ex æqua
li, the Time along B A ſhall be to the Time along B C as B A to B S,
or as C B to BS: But C B is to B S, as D B to B I: Therefore the
Propoſition is manifeſt:
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