83[6]
Therefore AT x AO is greater than o AO
Or AT greater than Ao.
Corollary II.
If the three given points be I, A, E;
and O falls between
A and I, ſo as to make AO x PE: IOE: : AL: LI, I ſay then O will fall
beyond L.
A and I, ſo as to make AO x PE: IOE: : AL: LI, I ſay then O will fall
beyond L.
For let us ſuppoſe that O and L coincide;
then by hypotbeſis AL:
LI:
:
AL x PE: IL x LE
AL x PE: IL x LE
And by the next following Lemma IV.
AL x IL:
IL x PE:
: AL:
LE
i. e. AL: PE: : AL: LE
i. e. AL: PE: : AL: LE
Hence PE is equal to LE, a part to the whole, which is manifeſtly abſurd.
LEMMA IV.
If it be as a line to a line ſo a rectangle to a rectangle;
then I ſay it will be
as the flrſt line into the breadth of the ſecond rectangle to the ſecond line into
the breadth of the firſt rectangle, ſo the length of the firſt rectangle to the
length of the ſecond.
as the flrſt line into the breadth of the ſecond rectangle to the ſecond line into
the breadth of the firſt rectangle, ſo the length of the firſt rectangle to the
length of the ſecond.
Suppoſition.
AE:
IO:
: UYN:
SRL.
Concluſion.
AE x RL:
IO x YN:
: UY:
SR.
Dem.
AE:
IO:
: AE x YN:
IO x YN:
: UYN:
SRL
And by Permutation AE x YN:
UYN:
: AE:
UY:
: IO x YN:
SRL
But SR:
AE:
: SRL:
: AE x RL
Therefore ex æquo perturbatè SR:
UY:
: IO x YN:
AE x RL
Q.
E.
D.
LEMMA V.
If a right line be cut in two points, I fay the rectangle under the alternate
ſegments is equal to that under the whole and the middle ſegment, together
with the rectangle under the extremes.
ſegments is equal to that under the whole and the middle ſegment, together
with the rectangle under the extremes.
Dem.
AI x IE + IO x IE = AO x IE.
Hence {AI x IE + IO x IE + AE x IO \\ i.
e.
AI x IE + AI x IO \\ i.
e.
AI x EO} = AO x IE + AE x IO.
Q.
E.
D.
N.
B.
Theſe two Lemmas ſave much Circumlocution and Tautology in
the two following Propoſitions, and indeed are highly uſeful in all caſes where
compound ratios are concerned.
the two following Propoſitions, and indeed are highly uſeful in all caſes where
compound ratios are concerned.