1if I ſhall ſhew you what proportion the Superficies of equall Cy
linders have to one another, we ſhall obtain our deſire. I ſay there
fore, that
linders have to one another, we ſhall obtain our deſire. I ſay there
fore, that
PROPOSITION.
The Superficies of Equal Cylinders, their Baſes being
ſubſtracted, are to one another in ſubduple proportion
of their lengths.
ſubſtracted, are to one another in ſubduple proportion
of their lengths.
Take two equall Cylinders, the heights of which let be A B,
and C D: and let the Line E be a Mean-proportional
between them. I ſay, the Superficies of the Cylinder A B,
the Baſes ſubſtracted, hath the ſame proportion to the Superficies
of the Cylinder C D, the Baſes in like manner ſubſtracted, as the
Line A B hath to the Line E, which is ſubduple of the proportion
of A B to C D. Cut the part of the Cylinder A B in F, and let the
height A F be equal to C D: And becauſe the Baſes of equal Cy
linders anſwer Reciprocally to their heights, the Circle, Baſe of
the Cylinder C D, to the Circle, Baſe of the
61[Figure 61]
Cylinder A B, ſhall be as the height B A to
D C: And becauſe Circles are to one ano
ther as the Squares of their Diameters, the
ſaid Squares ſhall have the ſame proportion,
that B A hath to C D: But as B A, is to
C D, ſo is the Square B A to the Square of
E: Therefore thoſe four Squares are Pro
portionals: And therefore their Sides ſhall
be Proportionals. And as the Line A B is to
E, ſo is the Diameter of the Circle C to the
Diameter of the Circle A: But as are the
Diameters, ſo are the Circumferences; and
as are the Circumferences, ſo likewiſe are the Superficies of Cylin
ders equal in Height. Therefore as the Line A B is to E, ſo is the
Superficies of the Cylinder C D to the Superficies of the Cylinder
A F. Becauſe therefore the height A F to the height A B, is as the
Superficies A F to the Superficies A B: And as is the height A B
to the Line E, ſo is the Superficies C D to the Superficies A F:
Therefore by Perturbation of Proportion as the height A F is to
E, ſo is the Superficies C D to the Superficies A B: And, by Con
verſion, as the Superficies of the Cylinder A B is to the Superficies
of the Cylinder C D, ſo is the Line E to the Line A F; that is, to
the Line C D: or as A B to E: Which is in ſubduple proportion
of A B to C D: Which is that which was to be proved.
and C D: and let the Line E be a Mean-proportional
between them. I ſay, the Superficies of the Cylinder A B,
the Baſes ſubſtracted, hath the ſame proportion to the Superficies
of the Cylinder C D, the Baſes in like manner ſubſtracted, as the
Line A B hath to the Line E, which is ſubduple of the proportion
of A B to C D. Cut the part of the Cylinder A B in F, and let the
height A F be equal to C D: And becauſe the Baſes of equal Cy
linders anſwer Reciprocally to their heights, the Circle, Baſe of
the Cylinder C D, to the Circle, Baſe of the
61[Figure 61]Cylinder A B, ſhall be as the height B A to
D C: And becauſe Circles are to one ano
ther as the Squares of their Diameters, the
ſaid Squares ſhall have the ſame proportion,
that B A hath to C D: But as B A, is to
C D, ſo is the Square B A to the Square of
E: Therefore thoſe four Squares are Pro
portionals: And therefore their Sides ſhall
be Proportionals. And as the Line A B is to
E, ſo is the Diameter of the Circle C to the
Diameter of the Circle A: But as are the
Diameters, ſo are the Circumferences; and
as are the Circumferences, ſo likewiſe are the Superficies of Cylin
ders equal in Height. Therefore as the Line A B is to E, ſo is the
Superficies of the Cylinder C D to the Superficies of the Cylinder
A F. Becauſe therefore the height A F to the height A B, is as the
Superficies A F to the Superficies A B: And as is the height A B
to the Line E, ſo is the Superficies C D to the Superficies A F:
Therefore by Perturbation of Proportion as the height A F is to
E, ſo is the Superficies C D to the Superficies A B: And, by Con
verſion, as the Superficies of the Cylinder A B is to the Superficies
of the Cylinder C D, ſo is the Line E to the Line A F; that is, to
the Line C D: or as A B to E: Which is in ſubduple proportion
of A B to C D: Which is that which was to be proved.