Salusbury, Thomas, Mathematical collections and translations (Tome I), 1667

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1and I will give you an anſwer. Tell me therefore, how much do
you think ſufficeth to make that motion ſwifter than this?
SIMP. I will ſay for example, that if that motion by the
gent were a million of times ſwifter than this by the ſecant, the
pen, yea, and the ſtone alſo would come to be extruded.
SALV. You ſay ſo, and ſay that which is falſe, onely for
want, not of Logick, Phyſicks, or Metaphyſicks, but of
try; for if you did but underſtand its firſt elements, you would
know, that from the centre of a circle a right line may be drawn
to meet the tangent, which interſecteth it in ſuch a manner, that
the part of the tangent between the contact and the ſecant, may
be one, two, or three millions of times greater than that part of
the ſecant which lieth between the tangent and the circumference,
and that the neerer and neerer the ſecant ſhall be to the contact,
this proportion ſhall grow greater and greater in infinitum; ſo
that it need not be feared, though the vertigo be ſwift, and the
motion downwards ſlow, that the pen or other lighter matter can
begin to riſe upwards, for that the inclination downwards always
exceedeth the velocity of the projection.
SAGR. I do not perfectly apprehend this buſineſſe.
SALV. I will give you a moſt univerſal yet very eaſie demon­

ſtration thereof.
Let a proportion be given between B A [in Fig.
3.] and C: And let B A be greater than C at pleaſure.
And let
there be deſcribed a circle, whoſe centre is D.
From which it is
required to draw a ſecant, in ſuch manner, that the tangent may
be in proportion to the ſaid ſecant, as B A to C.
Let A I be
ſuppoſed a third proportional to B A and C.
And as B I is to
I A, ſo let the diameter F E be to E G; and from the point G,
let there be drawn the tangent G H.
I ſay that all this is done as
was required; and as B A is to C, ſo is H G to G E.
And in
gard that as B I is to I A, ſo is F E to E G; therefore by
ſition, as B A is to A I; ſo ſhall F G be to G E.
And becauſe C
is the middle proportion between B A and A I; and G H is a
middle term between F G and G E; therefore, as B A is to C,
ſo ſhall F G be to G H; that is H G to G E, which was to be
demonſtrated.
A geometrical
demonſtration to
prove the
bility of extruſion
by means of the
terreſtrial vertigo.
SAGR. I apprehend this demonſtration; yet nevertheleſſe, I
am not left wholly without hæſitation; for I find certain
ſed ſcruples role to and again in my mind, which like thick and
dark clouds, permit me not to diſcern the cleerneſſe and neceſſity
of the concluſion with that perſpicuity, which is uſual in
matical Demonſtrations.
And that which I ſtick at is this. It is
true that the ſpaces between the tangent and the circumference do
gradually diminiſh in infinitum towards the contact; but it is alſo
true on the contrary, that the propenſion of the moveable to