1(which is equal to four Squares of A D:) Therefore four Rectan
gles E A D ſhall have greater proportion to the Square C D, than
the Square E D hath to the Square D C: Therefore four Rectan
gles E A D ſhall be greater than the Square E D: which is falſe,
for they are leſſe; becauſe the parts E A and A D of the Line E D
are not equal: Therefore the Line D B toucheth the Parabola in B,
and doth not cut it: Which was to be demonſtrated.
gles E A D ſhall have greater proportion to the Square C D, than
the Square E D hath to the Square D C: Therefore four Rectan
gles E A D ſhall be greater than the Square E D: which is falſe,
for they are leſſe; becauſe the parts E A and A D of the Line E D
are not equal: Therefore the Line D B toucheth the Parabola in B,
and doth not cut it: Which was to be demonſtrated.
SIMP. You proceed in your Demonſtrations too ſublimely,
and ſtill, as far as I can perceive, ſuppoſe that the Propoſitions of
Euclid are as familiar and ready with me, as the firſt Axioms them
ſelves, which is not ſo. And the impoſing upon me, juſt now, that
four Rectangles E A D are leſs than the Square D E becauſe the
parts E A and A D of the Line E D are not equal, doth not ſatisſie
me, but leaveth me in doubt.
and ſtill, as far as I can perceive, ſuppoſe that the Propoſitions of
Euclid are as familiar and ready with me, as the firſt Axioms them
ſelves, which is not ſo. And the impoſing upon me, juſt now, that
four Rectangles E A D are leſs than the Square D E becauſe the
parts E A and A D of the Line E D are not equal, doth not ſatisſie
me, but leaveth me in doubt.
SALV. The truth is, all the Mathematicians that are not vulgar
ſuppoſe that the Reader hath ready by heart the Elements of
Euclid: And here to ſupply your want, it ſhall ſuſfice to remember
you of a Propoſition in the ſecond Book, in which it is demonſtrated
that when a Line is cut into equal parts, and into unequal, the
Rectangle of the unequal parts is leſs than the Rectangle of the
equal, (that is, than the Square of the half) by ſo much as is the
Square of the Line comprized between the Sections. Whence it is
manifeſt, that the Square of the whole, which continueth four
Squares of the Half, is greater than four Rectangles of the unequal
parts. Now it is neceſſary that we bear in mind theſe two Propoſi
tions which have been demonſtrated, taken from the Conick Ele
ments, for the better underſtanding the things that follow in the
preſent Treatiſe: for of theſe two, and no more, the Author
makes uſe. Now we may reaſſume the Text to ſee in what manner
he doth demonſtrate his firſt Propoſition, in which he intendeth to
prove unto us, That the Line deſcribed by the Grave Moveable,
when it deſcends with a Motion compounded of the Equable
Horizontal, and of the Natural Deſcending is a Semiparabola.
ſuppoſe that the Reader hath ready by heart the Elements of
Euclid: And here to ſupply your want, it ſhall ſuſfice to remember
you of a Propoſition in the ſecond Book, in which it is demonſtrated
that when a Line is cut into equal parts, and into unequal, the
Rectangle of the unequal parts is leſs than the Rectangle of the
equal, (that is, than the Square of the half) by ſo much as is the
Square of the Line comprized between the Sections. Whence it is
manifeſt, that the Square of the whole, which continueth four
Squares of the Half, is greater than four Rectangles of the unequal
parts. Now it is neceſſary that we bear in mind theſe two Propoſi
tions which have been demonſtrated, taken from the Conick Ele
ments, for the better underſtanding the things that follow in the
preſent Treatiſe: for of theſe two, and no more, the Author
makes uſe. Now we may reaſſume the Text to ſee in what manner
he doth demonſtrate his firſt Propoſition, in which he intendeth to
prove unto us, That the Line deſcribed by the Grave Moveable,
when it deſcends with a Motion compounded of the Equable
Horizontal, and of the Natural Deſcending is a Semiparabola.
Suppoſe the Horizontal Line or Plane A B placed on high; upon
[or along] which let the Moveable paſſe with an Equable Motion out
of A unto B: and the ſupport of the Plane failing in B let there be
derived upon the Moveable from its own Gravity a Motion naturally
downwards according to the Perpendicular B N. Let the Line B E be
ſuppoſed applyed unto the Plane A B right out, as if it were the Efflux
or meaſure of the Time, on which at pleaſure note any equal parts of
Time, B C, C D, D E: And out of the points B C D E ſuppoſe Per
pendicular Lines to be let fall equidiſtant or parallel to B N: In the firſt
of which take any part C I, whoſe quadruple take in the following one
D F, nonuple E H, and ſo in the reſt that follow according to the propor-
[or along] which let the Moveable paſſe with an Equable Motion out
of A unto B: and the ſupport of the Plane failing in B let there be
derived upon the Moveable from its own Gravity a Motion naturally
downwards according to the Perpendicular B N. Let the Line B E be
ſuppoſed applyed unto the Plane A B right out, as if it were the Efflux
or meaſure of the Time, on which at pleaſure note any equal parts of
Time, B C, C D, D E: And out of the points B C D E ſuppoſe Per
pendicular Lines to be let fall equidiſtant or parallel to B N: In the firſt
of which take any part C I, whoſe quadruple take in the following one
D F, nonuple E H, and ſo in the reſt that follow according to the propor-