THEOR. VI. PROP. VI.
If from the higheſt or loweſt part of a Circle,
erect upon the Horizon, certain Planes be
drawn inclined towards the Circumference,
the Times of the Deſcents along the ſame
ſhall be equal.
erect upon the Horizon, certain Planes be
drawn inclined towards the Circumference,
the Times of the Deſcents along the ſame
ſhall be equal.
Let the Circle be erect upon the Horizon G H, whoſe Diameter
recited upon the loweſt point, that is upon the contact with the
Horizon, let be F A, and from the higheſt point A let certain
Planes A B and A C incline towards the Circumference: I ſay that the
Times of the Deſcents along the ſame are equal. Let B D and C E be
two Perpendiculars let fall unto the Diameter; and let A I be a Mean
Proportional between the Altitudes
93[Figure 93]
of the Planes E A and A D. And
becauſe the Rectangles F A E and
F A D are equal to the Squares of
A C and A B; And alſo becauſe
that as the Rectangle F A E, is to
the Rectangle F A D, ſo is E A to
A D. Therefore as the Square of
C A is to the Square of B A,
ſo is the Line E A to the Line
A D. But as the Line E A is to
D A, ſo is the Square of I A to the Square of A D: Therefore
the Squares of the Lines C A and A B are to each other as the Squares
of the Lines I A and A D: And therefore as the Line C A is to A B,
ſo is I A to A D: But in the precedent Propoſition it hath been demon
ſtrated that the proportion of the Time of the Deſcent along A C to the
Time of the Deſcent by A B, is compounded of the proportions of C A
to A B, and of D A to A I, which is the ſame with the proportion of
B A to A C: Therefore the proportion of the Time of the Deſcent along
A C, to the Time of the Deſcent along A B, is compounded of the pro
portions of C A to A B, and of B A to A C: Therefore the proporti
on of thoſe Times is a proportion of equality: Therefore the Propoſition
is evident.
recited upon the loweſt point, that is upon the contact with the
Horizon, let be F A, and from the higheſt point A let certain
Planes A B and A C incline towards the Circumference: I ſay that the
Times of the Deſcents along the ſame are equal. Let B D and C E be
two Perpendiculars let fall unto the Diameter; and let A I be a Mean
Proportional between the Altitudes
93[Figure 93]of the Planes E A and A D. And
becauſe the Rectangles F A E and
F A D are equal to the Squares of
A C and A B; And alſo becauſe
that as the Rectangle F A E, is to
the Rectangle F A D, ſo is E A to
A D. Therefore as the Square of
C A is to the Square of B A,
ſo is the Line E A to the Line
A D. But as the Line E A is to
D A, ſo is the Square of I A to the Square of A D: Therefore
the Squares of the Lines C A and A B are to each other as the Squares
of the Lines I A and A D: And therefore as the Line C A is to A B,
ſo is I A to A D: But in the precedent Propoſition it hath been demon
ſtrated that the proportion of the Time of the Deſcent along A C to the
Time of the Deſcent by A B, is compounded of the proportions of C A
to A B, and of D A to A I, which is the ſame with the proportion of
B A to A C: Therefore the proportion of the Time of the Deſcent along
A C, to the Time of the Deſcent along A B, is compounded of the pro
portions of C A to A B, and of B A to A C: Therefore the proporti
on of thoſe Times is a proportion of equality: Therefore the Propoſition
is evident.
The ſame is another way demonſtrated from the Mechanicks: Name
ly that in the enſuing Figure the Moveable paſſeth in equal Times along
C A and D A. For let B A be equal to the ſaid D A, ond let fall the
Perpendiculars B E and D F: It is manifeſt by the Elements of the
ly that in the enſuing Figure the Moveable paſſeth in equal Times along
C A and D A. For let B A be equal to the ſaid D A, ond let fall the
Perpendiculars B E and D F: It is manifeſt by the Elements of the