THEOR. XVII. PROP. XXVII.
If a Moveable deſcend along unequal Planes,
whoſe Elevation is the ſame, the Space that
ſhall be paſt along the lower part of the longeſt
in a Time equal to that in which the whole
ſhorter Plane is paſſed, is equal to the Space
that is compounded of the ſaid ſhorter Plane
and of the part to which that ſhorter Plane
hath the ſame Proportion that the longer
Plane hath to the Exceſſe by which the longeſt
exceedeth the ſhorteſt.
whoſe Elevation is the ſame, the Space that
ſhall be paſt along the lower part of the longeſt
in a Time equal to that in which the whole
ſhorter Plane is paſſed, is equal to the Space
that is compounded of the ſaid ſhorter Plane
and of the part to which that ſhorter Plane
hath the ſame Proportion that the longer
Plane hath to the Exceſſe by which the longeſt
exceedeth the ſhorteſt.
Let A C be the longer Plane, and A B the ſhorter, whoſe Elevation
A D is the ſame; and in the lower part of A C take the Space
C E, equal to the ſaid A B; and as C A is to A E, (that is to
the exceſſe of the Plane C A above A B) ſo let C E be to E F. I ſay,
that the Space F C is that which is paſt after the Deſcent out of A in
a Time equal to the Time of
125[Figure 125]
the Deſcent along A B. For
the whole C A, being to the
whole A E, as the part taken
away C E is to the part taken
away E F, therefore the re
maining part E A ſhall be to
the remaining part A F, as the
whole C A is to the whole A E: Therefore the three Spaces C A,
A E, and A F are three Continual proportionals. And if the Time
along A B be ſuppoſed to be as A B, the Time along A C ſhall be as
A C, and the Time along A F ſhall be as A E, and along the remain
ing part F C ſhall be as E C: But E C is equal to the ſaid A B: There
fore the Propoſition is manifeſt.
A D is the ſame; and in the lower part of A C take the Space
C E, equal to the ſaid A B; and as C A is to A E, (that is to
the exceſſe of the Plane C A above A B) ſo let C E be to E F. I ſay,
that the Space F C is that which is paſt after the Deſcent out of A in
a Time equal to the Time of
125[Figure 125]the Deſcent along A B. For
the whole C A, being to the
whole A E, as the part taken
away C E is to the part taken
away E F, therefore the re
maining part E A ſhall be to
the remaining part A F, as the
whole C A is to the whole A E: Therefore the three Spaces C A,
A E, and A F are three Continual proportionals. And if the Time
along A B be ſuppoſed to be as A B, the Time along A C ſhall be as
A C, and the Time along A F ſhall be as A E, and along the remain
ing part F C ſhall be as E C: But E C is equal to the ſaid A B: There
fore the Propoſition is manifeſt.
THEOR. XVIII. PROP. XXVIII.
Let the Horizontal Line A G be Tangent to a Circle, and from the
point of Contact let A B be the Diameter, and A E B two Chords
at pleaſure: We are to aſſign the proportion of the Time of the
Fall along A B to the Time of the Deſcent along both the Chords
A E B. Let B E be continued out till it meet the Tangent in G, and
point of Contact let A B be the Diameter, and A E B two Chords
at pleaſure: We are to aſſign the proportion of the Time of the
Fall along A B to the Time of the Deſcent along both the Chords
A E B. Let B E be continued out till it meet the Tangent in G, and