PROP. XVI. PROBL. VI.
A Trunk or Hollow Cane being given, a Solid Cylinder
may be found equal to it.
may be found equal to it.
This Operation is very eaſie. For let the Line A B, be the Dia
meter of the Cane, and C D the Diameter of the Hollow or
Cavity. Let the Line A E be ſet off upon the greater Circle
equal to the Diameter C D, and conjoyn E B. And becauſe in
74[Figure 74]
the Semicircle A E B the Angle E is Right
Angle, the Circle whoſe Diameter is A B
ſhall be equall to the two Circles of the Di
ameters A E and E B: But A E is the Dia
meter of the Hollow of the Cane: Therefore
the Circle whoſe Diameter is E B, ſhall be
equal to the Ring A C B D: And therefore
the ſolid Cylinder, the Circle of whoſe Baſe
hath the Diameter E B ſhall be equal to the
Cane, they being of the ſame length. This demonſtrated, we may
preſently be able
meter of the Cane, and C D the Diameter of the Hollow or
Cavity. Let the Line A E be ſet off upon the greater Circle
equal to the Diameter C D, and conjoyn E B. And becauſe in
74[Figure 74]the Semicircle A E B the Angle E is Right
Angle, the Circle whoſe Diameter is A B
ſhall be equall to the two Circles of the Di
ameters A E and E B: But A E is the Dia
meter of the Hollow of the Cane: Therefore
the Circle whoſe Diameter is E B, ſhall be
equal to the Ring A C B D: And therefore
the ſolid Cylinder, the Circle of whoſe Baſe
hath the Diameter E B ſhall be equal to the
Cane, they being of the ſame length. This demonſtrated, we may
preſently be able
PROP. XVII. PROBL. VII.
To find what proportion is betwixt the Reſiſtances of
any whatſoever Cane and Cylinder, their lengths be
ing equal.
any whatſoever Cane and Cylinder, their lengths be
ing equal.
LET the Cane A B E, and the Cylinder R S M, be of equal
length: it is required to find what proportion the Reſiſtances
have to each other. By the precedent let the Cylinder I L N
be found equal to the Cane, and of the ſame length; and to the
Lines I L and R S (Diameters of the Baſes of the Cylinders I N and
75[Figure 75]
R M) let the Line V be a fourth
Proportional. I ſay, the Reſiſtance
of the Cane A E is to the Reſi
ſtance of the Cylinder R M, as the
Line A B is to V. For the Cane
A E being equal to, and of the
ſame length with the Cylinder
I N, the Reſiſtance of the Cane
ſhall be to the Reſiſtance of the
Cylinder, as the Line A B is to I L:
But the Reſiſtance of the Cylinder I N is to the Reſiſtance of the
Cylinder R M, as the Cube I L is to the Cube R S; that is, as the
Line I L to V: Therefore, ex æquali, the Reſiſtance of the Cane
A E hath the ſame proportion to the Reſiſtance of the Cylinder
R M, that the Line A B hath to V: Which is that that was ſought.
length: it is required to find what proportion the Reſiſtances
have to each other. By the precedent let the Cylinder I L N
be found equal to the Cane, and of the ſame length; and to the
Lines I L and R S (Diameters of the Baſes of the Cylinders I N and
75[Figure 75]R M) let the Line V be a fourth
Proportional. I ſay, the Reſiſtance
of the Cane A E is to the Reſi
ſtance of the Cylinder R M, as the
Line A B is to V. For the Cane
A E being equal to, and of the
ſame length with the Cylinder
I N, the Reſiſtance of the Cane
ſhall be to the Reſiſtance of the
Cylinder, as the Line A B is to I L:
But the Reſiſtance of the Cylinder I N is to the Reſiſtance of the
Cylinder R M, as the Cube I L is to the Cube R S; that is, as the
Line I L to V: Therefore, ex æquali, the Reſiſtance of the Cane
A E hath the ſame proportion to the Reſiſtance of the Cylinder
R M, that the Line A B hath to V: Which is that that was ſought.
The End of the Second Dialogue.
zoom in
zoom out
zoom area
full page
page width
set mark
remove mark
get reference
digilib