1applyed to the Line B G will equal the Reſiſtance D A, will like
wiſe equal the Reſiſtance C O. And the ſame may be demonſtra
ted, if one cut the Solid in any other place: therefore that Parabo
lical Solid is throughout of equal Reſiſtance. In the next place,
that cutting the Priſme according to the Parabolical Line F N B,
the third part of it is taken away, appeareth, For that the Semi
Parabola F N B A and the Rectangle F B are Baſes of two Solids
contained between two parallel Planes, that is, between the Rect
angles F B and D G, whereby they retain the ſame Proportion, as
thoſe their Baſes: But the Rectangle F B is Seſquialter to the Se
miparabola F N B A: Therefore cutting the Priſine according to
the Parabolick Line, we take away the third part of it. Hence we
ſee, that Beams may be made with the diminution of their Weight
more than thirty three in the hundred, without diminiſhing their
Strength in the leaſt; which in great Ships, in particular, for bea
ring the Decks may be of no ſmall benefit; for that in ſuch kind
of Fabricks Lightneſſe is of infinite importance.
wiſe equal the Reſiſtance C O. And the ſame may be demonſtra
ted, if one cut the Solid in any other place: therefore that Parabo
lical Solid is throughout of equal Reſiſtance. In the next place,
that cutting the Priſme according to the Parabolical Line F N B,
the third part of it is taken away, appeareth, For that the Semi
Parabola F N B A and the Rectangle F B are Baſes of two Solids
contained between two parallel Planes, that is, between the Rect
angles F B and D G, whereby they retain the ſame Proportion, as
thoſe their Baſes: But the Rectangle F B is Seſquialter to the Se
miparabola F N B A: Therefore cutting the Priſine according to
the Parabolick Line, we take away the third part of it. Hence we
ſee, that Beams may be made with the diminution of their Weight
more than thirty three in the hundred, without diminiſhing their
Strength in the leaſt; which in great Ships, in particular, for bea
ring the Decks may be of no ſmall benefit; for that in ſuch kind
of Fabricks Lightneſſe is of infinite importance.
SAGR. The Commodities are ſo many, that it would be tedi
ous, if not impoſſible, to mention them all. But I, laying aſide
theſe, would more gladly underſtand that the alleviation is made
according to the aſſigned proportions. That the Section, according
to the Diagonal Line, cuts away half of the weight I very well
know: but that the other Section according to the Parabolical Line
takes away the third part of the Priſme I can believe upon the
word of Salviatus, who evermore ſpeaks the truth, but in this
Caſe Science would better pleaſe me than Faith.
ous, if not impoſſible, to mention them all. But I, laying aſide
theſe, would more gladly underſtand that the alleviation is made
according to the aſſigned proportions. That the Section, according
to the Diagonal Line, cuts away half of the weight I very well
know: but that the other Section according to the Parabolical Line
takes away the third part of the Priſme I can believe upon the
word of Salviatus, who evermore ſpeaks the truth, but in this
Caſe Science would better pleaſe me than Faith.
SALV. I ſee then that you would have the Demonſtration,
whether or no it be true, that the exceſſe of the Priſme over and
above this, which for this time we will call a Parabolical Solid, is
the third part of the whole Priſme. I am certain that I have for
merly demoſtrated it; I will try now whether I can put the
Demonſtration together again: to which purpoſe I do remember
that I made uſe of a Certain Lemma of Archimedes, inſerted by
him in his Book de Spiralibus, and it is this:
whether or no it be true, that the exceſſe of the Priſme over and
above this, which for this time we will call a Parabolical Solid, is
the third part of the whole Priſme. I am certain that I have for
merly demoſtrated it; I will try now whether I can put the
Demonſtration together again: to which purpoſe I do remember
that I made uſe of a Certain Lemma of Archimedes, inſerted by
him in his Book de Spiralibus, and it is this:
LEMMA II.
If any number of Lines at pleaſure ſhall exceed one another equal
ly, and the exceſſes be equal to the leaſt of them, and there be as
many more, each of them equal to the greateſt; the Squares of all
theſe ſhall be leſſe than the triple of the Squares of thoſe that
exceed one another: but they ſhall be more than triple to thoſe
others that remain, the Square of the greateſt being ſub
ſtracted.
ly, and the exceſſes be equal to the leaſt of them, and there be as
many more, each of them equal to the greateſt; the Squares of all
theſe ſhall be leſſe than the triple of the Squares of thoſe that
exceed one another: but they ſhall be more than triple to thoſe
others that remain, the Square of the greateſt being ſub
ſtracted.