54 
TRANSACTIONS OF THE TEXAS ACADEMY OF SCIENCE. 
string be tied loosely about the inner conic and drawn tight with a 
tracing point, the point describes a conic byconfocal with the first. 
1° contains Fagnani’s theorem as a particular case, for if the tan¬ 
gents be drawn from the points where the major and minor axes cut 
the curve, the elliptic quadrant will be cut up into parts whose differ¬ 
ence is expressible by a constructable line. 
2 ° . The reciprocal of ( b ) is: When tw r o byconfocal conics cut each 
other, if tangents be drawn from any point on one to the other their 
difference is equal to the difference of the arcs between their intersec¬ 
tion and the points where the tangents touch. 
Cor. If tangents be drawn at the point above the major and minor 
axes cut the curve, and a byconfocal conic be traced through their in¬ 
tersection, it will intersect the original conic in a point (Fagnani’s), 
which divides the quadrant into two parts, whose difference, in piano, 
is equal to the difference between the axes. 
24. By means of 2 ° , it is also possible to prove a theorem of Lan- 
den’s. (See Williamson’s Integral Calculus.) 
In figure 14, DS is a tangent at 
the vertex, and CB is a tangent from 
the center of the two conics. CB is 
a quadrant, and by 2° DB—DS= 
AB—AS, or adding CD and 2AS. 
CB-BS=DS-2AS+CD. This re¬ 
sult is mainly interesting on account 
of its interpretation in piano. 
THE PLANE. 
If the sphere increases without limit, the tri-rectangular triangle 
becomes indefinitely large, while the quadrant becomes immeasurably 
long. This last is evident, because the reciprocal of a tri-rectangular 
triangle is a tri-rectangular triangle, hence from the relation existing 
between the sides and angles (area) of two reciprocal figures, the in¬ 
finitude of one implies the infinitude of the other. 
Since the area of a polygon is equal to the spherical excess, multiplied 
by the tri-iectangular triangle, if a polygon of finite extent be drawn on 
a sphere, whose quadrant is becoming indefinitely long, its spherical 
excess will have to approach zero as a limit, and over a finite area the 
assumption of parallels holds good, and the rules of planimetry apply. 
It is thus, by the measurement of the spherical excess of astronomical 
triangles, that attempts have been made to determine the curvature of 
our cosmical space. On the other hand, the rules of planimetry and 
similarity are applicable, on a finite sphere, to indefinitely small areas, 
and the rigorous proof of Graves’ theorem, etc., requires it to be proven 
