acte
proportion betwene two thynges, and yet can not name nor attayne
the precise quantitee of those two thynges, As for exaumple, If
two squares be sette foorthe, whereof the one containeth in it
fiue square feete, and the other contayneth fiue and fortie
foote, of like square feete, I am not able to tell, no nor yet
anye manne liuyng, what is the precyse measure of the sides of
any of those .ij. squares, and yet I can proue by vnfallible
reason, that their sides be in a triple proportion, that is to
saie, that the side of the greater square (whiche containeth
.xlv. foote) is three tymes so long iuste as the side of the
lesser square, that includeth but fiue foote. But this seemeth
to be spoken out of ceason in this place, therfore I will omitte
it now, reseruyng the exacter declaration therof to a more
conuenient place and time, and will procede with the residew of
the Theoremes appointed for this boke.

_The .lxi. Theoreme._

If a right line be drawen at any end of a diameter in
perpendicular forme, and do make a right angle with the
diameter, that right line shall light without the circle,
and yet so iointly knitte to it, that it is not possible to
draw any other right line betwene that saide line and the
circumference of the circle. And the angle that is made in
the semicircle is greater then any sharpe angle that may be
made of right lines, but the other angle without, is lesser
then any that can be made of right lines.

_Example._

[Illustration]

In this circle A.B.C, the diameter is A.C, the perpendicular
line, which maketh a right angle with the diameter, is C.A,
whiche line falleth without the circle, and yet ioyneth so
exactly vnto it, that it is not possible to draw an other right
line betwene the circumference of the circle and it, whiche
thyng is so plainly seene of the eye, that it needeth no farther
declaracion. For euery man wil easily consent, that betwene the
croked line A.F, (whiche is a parte of the circumference of the
circle) and A.E (which is the said perpendicular line) there can
none other line bee drawen in that place where they make the
angle. Nowe for the residue of the theoreme. The angle D.A.B,
which is made in the semicircle, is greater then anye sharpe
angle that may bee made of ryghte lines. and yet is it a sharpe
angle also, in as much as it is lesser then a right angle, which
is the angle E.A.D, and the residue of that right angle, which
lieth without