10248An ESSAY
might be done) the Point on the Semicircle
h z m, as far as the Parallels (as 9 m) are uſeful:
For when theſe Parallels are uſeleſs, the Point
q will fall beyond the Point m: But then the
Perſpective of the Torus is entirely drawn alrea-
dy, if thoſe Parallels were firſt begun to be drawn
near to 6 3 z, and the others continually going
from it.
h z m, as far as the Parallels (as 9 m) are uſeful:
For when theſe Parallels are uſeleſs, the Point
q will fall beyond the Point m: But then the
Perſpective of the Torus is entirely drawn alrea-
dy, if thoſe Parallels were firſt begun to be drawn
near to 6 3 z, and the others continually going
from it.
In order to demonſtrate this Problem, the fol-
lowing Lemma is neceſſary.
lowing Lemma is neceſſary.
Lemma.
65.
If two Circles C D H E and D E F L cut
11Fig. 34. each other, thro’ whoſe Centers C and B the Line
C L paſſes, and D E joyns their Interſections;
then, if the Radius A C or A H be called a, and
B F or BL, b, and the Diſtance A B between the
two Centers c, I ſay A G is equal to {bb—aa/ec}—{1/2}C.
11Fig. 34. each other, thro’ whoſe Centers C and B the Line
C L paſſes, and D E joyns their Interſections;
then, if the Radius A C or A H be called a, and
B F or BL, b, and the Diſtance A B between the
two Centers c, I ſay A G is equal to {bb—aa/ec}—{1/2}C.
Demonstration.
Let us call A G, x, and G D or G E, y.
Then by the Property of the Circle, if y be conceiv’d
as an Ordinate of the Circle, C D H; yy=aa—xx.
And if it be likewiſe conſider’d as an Ordinate of
the Circle F D L, yy=bb—cc—2cx—xx: Whence
aa—xx=bb—cc—2cx—xx, and ſo 2cx=bb
—aa—cc; and dividing each Side of this laſt Equa-
tion by 2c, we have a x={bb—aa/2c}{1/2} c. Which was
to be Demonſtrated.
Then by the Property of the Circle, if y be conceiv’d
as an Ordinate of the Circle, C D H; yy=aa—xx.
And if it be likewiſe conſider’d as an Ordinate of
the Circle F D L, yy=bb—cc—2cx—xx: Whence
aa—xx=bb—cc—2cx—xx, and ſo 2cx=bb
—aa—cc; and dividing each Side of this laſt Equa-
tion by 2c, we have a x={bb—aa/2c}{1/2} c. Which was
to be Demonſtrated.