1pendicular to the Diameter A I: Wherefore alſo A I toucheth the Cir
cle B O E in I. And, becauſe the Angle A I C is greater than the An
gle A B C, as inſiſting on a larger Periphery: Therefore the Angle
S I N ſhall be alſo greater than the ſame A B C: Therefore the Portion
I E S is greater than the Portion B O; and the Line C S, nearer to the
Center, greater than C B: Therefore alſo C O is greater than C I;
for that S C is to C B, as O C is to C I.
cle B O E in I. And, becauſe the Angle A I C is greater than the An
gle A B C, as inſiſting on a larger Periphery: Therefore the Angle
S I N ſhall be alſo greater than the ſame A B C: Therefore the Portion
I E S is greater than the Portion B O; and the Line C S, nearer to the
Center, greater than C B: Therefore alſo C O is greater than C I;
for that S C is to C B, as O C is to C I.
And the ſame alſo would happen to be greater, if (as in the other
Figure) the Quadrant B I C were
139[Figure 139]
leſſer: For the Perpendicular D B
will cut the Circle C I B: Wherefore
D I alſo is equal to the ſaid D B; and
the Angle D I A ſhall be Obtuſe, and
therefore A I N will alſo cut B I N:
And becauſe the Angle A B C is leſſe
than the Angle A I C, which is equal
to S I N; and this now is leſſe than that
which would be made at the Contact in
I by the Line S I: Therefore the Porti
on S E I is much greater than the Por
tion B O: Wherefore, &c. Which was
to be demonſtrated.
Figure) the Quadrant B I C were
139[Figure 139]leſſer: For the Perpendicular D B
will cut the Circle C I B: Wherefore
D I alſo is equal to the ſaid D B; and
the Angle D I A ſhall be Obtuſe, and
therefore A I N will alſo cut B I N:
And becauſe the Angle A B C is leſſe
than the Angle A I C, which is equal
to S I N; and this now is leſſe than that
which would be made at the Contact in
I by the Line S I: Therefore the Porti
on S E I is much greater than the Por
tion B O: Wherefore, &c. Which was
to be demonſtrated.
THEOR. XXII. PROP. XXXVI.
If from the loweſt point of a Circle erect unto
the Horizon a Plane ſhould be elevated ſub
tending a Circumference not greater than a
Quadrant, from whoſe Terms two other
Planes are Inflected to any point of the Cir
cumference, the Deſcent along both the Infle
cted Planes would be performed in a ſhorter
Time than along the former elevated Plane
alone, or than along but one of the other two,
namely, along the lower.
the Horizon a Plane ſhould be elevated ſub
tending a Circumference not greater than a
Quadrant, from whoſe Terms two other
Planes are Inflected to any point of the Cir
cumference, the Deſcent along both the Infle
cted Planes would be performed in a ſhorter
Time than along the former elevated Plane
alone, or than along but one of the other two,
namely, along the lower.
Let C B D be the Circumference not greater than a Quadrant of a
Circle erect unto the Horizon on the lower point C, in which let
C D be an elevated Plane; and let two Planes be inflected from the
Terms D and C to any point in the Circumference taken at pleaſure,
as B. I ſay, that the Time of the Deſcent along both thoſe Planes D B C
is ſhorter than the Time of the Deſcent along the ſole Plane D C, or
along the other only B C ex quiete in B. Let the Horizontal Line M D A
Circle erect unto the Horizon on the lower point C, in which let
C D be an elevated Plane; and let two Planes be inflected from the
Terms D and C to any point in the Circumference taken at pleaſure,
as B. I ſay, that the Time of the Deſcent along both thoſe Planes D B C
is ſhorter than the Time of the Deſcent along the ſole Plane D C, or
along the other only B C ex quiete in B. Let the Horizontal Line M D A