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

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1deſcending groweth leſs & leſs in it, the nearer it is to the firſt term
of its deſcent; that is, to the ſtate of reſt; as is manifeſt from that
which you declare unto us, demonſtrating that the deſcending grave
body departing from reſt, ought to paſſe thorow all the degrees of
tardity comprehended between the ſaid reſt, & any aſſigned degree
of velocity, the which grow leſs and leſs in infinitum. To which may
be added, that the ſaid velocity and propenſion to motion, doth for
another reaſon diminiſh to infinity; and it is becauſe the gravity of
the ſaid moveable may infinitely diminiſh.
So that the cauſes which
diminiſh the propenſion of aſcending, and conſequently favour
the projection, are two; that is, the levity of the moveable, and its
vicinity to the ſtate of reſt; both which are augmentable in infinit.
and theſe two on the contrary being to contract but with one ſole
cauſe of making the projection, I cannot conceive how it alone,
though it alſo do admit of infinite augmentation, ſhould be able to
remain invincible againſt the union & confederacy of the others, w^{ch}
are two, and are in like manner capable of infinite augmentation.
SALV. This is a doubt worthy of Sagredus; and to explain it ſo as
that we may more cleerly apprehend it, for that you ſay that you
your ſelf have but a confuſed Idea of it, we will diſtinguiſh of the
ſame by reducing it into figure; which may alſo perhaps afford us
ſome caſe in reſolving the ſame.
Let us therefore [in Fig. 4.] draw
a perpendicular line towards the centre, and let it be AC, and to it
at right angles let there be drawn the Horizontal line A B, upon
which the motion of the projection ought to be made; now the
ject would continue to move along the ſame with an even motion, if
ſo be its gravity did not incline it downwards.
Let us ſuppoſe from
the point A a right line to be drawn, that may make any angle at
pleaſure with the line A B; which let be A E, and upon AB let us
mark ſome equal ſpaces AF, FH, HK, and from them let us let fall
the perpendiculars FG, HI, K L, as far as AE.
And becauſe, as al
ready hath been ſaid, the deſcending grave body departing from reſt,
goeth from time to time acquiring a greater degree of velocity,
according as the ſaid time doth ſucceſſively encreaſe; we may
ceive the ſpaces AF, FH, HK, to repreſent unto us equal times; and
the perpendiculars FG, HI, KL, degrees of velocity acquired in the
ſaid times; ſo that the degree of velocity acquired in the whole time
A K, is as the line K L, in reſpect to the degree H I, acquired in the
time AH, and the degree FG in the time AF; the which degrees KL,
HI, FG, are (as is manifeſt) the ſame in proportion, as the times K A,
HA, F A, and if other perpendiculars were drawn from the points
marked at pleaſure in the line F A, one might ſucceſſively find
grees leſſe and leſſe in infinitum, proceeding towards the point A,
repreſenting the firſt inſtant of time, and the firſt ſtate of reſt.
And
this retreat towards A, repreſenteth the firſt propenſion to the