j -y 
SPII 
or expanse, which invests our globe, and in 
wliich the heavenly bodies appear to be fix- 
ed, and at an equal distance from the eye. 
Tlie better to determine tlie places of the 
heavenly bodies in the sphere, several cir- 
cles are supposed to be described on the 
surface thereof, hence called the circles of 
the sphere : of these, some are called great 
circles, as the equinoctial, ecliptic, meri- 
dian, &c. and others, small circles, as the 
tropics, parallels, &c. See each under its 
proper article. 
SPHERICS, is that part of geometry 
which treats of the position and mensura- 
tion of arches of circles, described on the ' 
surface of a sphere. See Sphere. 
SPHEROID, in geometry, a solid, ap- 
proaching to the figure of a sphere. 
The spheroid is generated by the entire 
revolution of a semi-ellipsis about its axis. 
Tims, if the semi-ellipsis A H F B (Plate 
XIV. Miscel. fig. 6,) be supposed to re- 
volve round its transverse axis A B, it will 
generate the oblong spheroid A H F B C. 
Now as all circles are as the squares de- 
scribed upon their radii ; that isj the circle 
of the radius E H, is to the circle of the ra- 
dius EG, as C F- to C D^, becaiise E H : 
E G : : C F ; C D, and since it is so every 
where, all the circles described with the re- 
spective radii EH, (that is, the spheroid 
made by the rotation of the semi ellipsis A 
F B about the a.xis A B) will be to all the 
circles described by the respective radii EG, 
(that is, the sphere described by the ro- 
tation of the semi-circle A D B on the axis 
A B) as F to C that is, as the sphe- 
roid is to the sphere on the same axis, so is 
the other axis of the generating ellipsis to 
the square of the diameter or axis of the 
sphere: and this holds whether the sphe- 
roid beiformed by a revolution around the 
greater or lesser axis. 
Hence it appears, that the half of the 
spheroid, formed by the rotation of the 
space A H F C, around the axis A C, is dou- 
ble of the cone generated by the triangle 
A F C, about the same axis. Hence, also, 
is evident the measure of the segments of 
the spheroid, cut by planes perpendicular 
to the axis: for the segment of the sphe- 
roid, made by the rotation of the space 
A N H E round the axis A E, is to the seg- 
ment of the sphere, having the same axis A 
C, and made by the rotation of the seg- 
ment of the circle A M G E, as C F^ to 
C D^. But the measuie’of this solid may be 
found with less ttwible by this analogy ; 
j;is. as B E: A C -f- E B ; : so is the cone 
SI’H 
generated by the rotation of the triangle 
A HE round the axis A E : to the segment 
of the sphere made by the rotation of the 
space A N H E round the same axis A E, 
as is demonstrated by Archimedes of co- 
noids and spheroids, prop. 34. This agrees 
as well to the oblate as to the oblong sphe- 
roid, A spheroid is also equal to two thirds 
of its circumscribing Cylinder. As to the 
superficies of a spheroid, M. Huygens 
gives the two following constructions in Ids 
Horolog. Oscill.' For describing a circle 
equal to the superficies of an oblong and 
prolate spheroid: 1. Let an oblong sphe- 
roid be generated by the rotation of the 
ellipsis • A D B E, (fig. 7) about its trans- 
verse axis A B, and let D E be its conju- 
gate; make D F equal to C B, or let F be 
one of the foci, and draw B G parallel to F 
D, and about the point G, with the radius 
B G, describe an arch, B H A, of a circle; 
then between the semi-conjugate C D, and 
a right line equal to 13 E -j- the arch A H 
B, find a mean proportional, and that will 
be the radius of a circle equal to the super- 
ficies of the oblong spheroid. S. Let a 
prolate spheroid be generated by the rota- 
tion of the ellipsis A D B E (fig. 8) about 
its conjugate axis A B. Let F be one of 
the foci, and bisect C F in G, and let A G B 
be the curve of the common parabola 
whose base is the conjugate diameter A B, 
and axis C G. Then if between the trans- 
verse axis D E, and a right line equal to 
the curve A G H of the parabola, a mean 
proportional be taken, the same will be the 
radius of a circle equal to the surface of 
that prolate spheroid. 
SPHEX, in natural history, a genus of 
insects of the order Hymenoptera. Month 
with an entire jaw, the mandibles horny, 
incurved, toothed; tip horny, membrana- 
ceous at the tip ; four feelers ; antenn® 
with about ten articulations ; wings, in each 
sex, plane, incumbent, and not folded; sting 
pungent and concealed within the abdo- 
men. There are about one hundred and fifty 
species, divided into sections. A. antenn®, 
setaceous ; and entire lip and no tongue. 
B. antenn® filiform ; lip einarginate, with 
a bristle on each side; tongue inflected, 
trifid. The insects of this genus are the 
most savage and rapacious of this class of 
beings : they attack whatever comes in 
their way, and by means of their poisonous 
sting overcome and devour others far be- 
yond their own size. Those belonging to 
section B are found chiefly on umbellate 
plants ; the larva is without feet, soft, and 
