ON CONOIDS AND SPHEROIDS 
57 
9450 : 23409 
ints marked 
tio a 2 : c and 
iplying each 
case by 
aken as the 
, or of the 
ver, be sure 
ntmla (a ± h) 2 
g the square 
iccording to 
ation to the 
re, in Archi- 
,o Dositheus, 
to the solid 
iinitions and 
1) the right- 
htuse-angled 
le spheroids 
ellipse about 
described by 
ieter’ (minor 
i axis of the 
fluent being- 
solid which 
ntre is only 
rresponds to 
e ‘ vertex of 
ion of what 
ction of the 
hyperbola), 
if the hyper- 
ber, but (the 
egment is in 
vertex of the 
in the case 
of the hyperboloid the portion within the solid of the line 
joining the vertex of the enveloping cone to the vertex of 
the segment and produced, and in the case of the spheroids the 
line joining the points of contact of the two tangent planes 
parallel to the base of the segment. Definitions are added of 
a ‘ segment of a cone ’ (the figure cut off towards the vertex by 
an elliptical, not circular, section of the cone) and a ‘ frustum 
of a cylinder’ (cut off by two parallel elliptical sections). 
Props. 1 to 18 with a Lemma at the beginning are preliminary 
to the main subject of the treatise. The Lemma and Props. 1, 2 
are general propositions needed afterwards. They include 
propositions in summation, 
2 [a + 2a + 3a+ ... + na] > n .na > 2 [a + 2a + ... + (n— 1)«} 
(Lemma) 
(this is clear from S n = \n (n + l)a) ; 
(n + 1) (na) 2 + a(a + 2a + 3a+ ... + na) 
= 3 [a 2 + (2a) 2 + (3a) 2 + ... + (na) 2 } ; 
(Lemma to Prop. 2) 
whence (Cor.) 
3 [a 2 + (2a) 2 + (3a) 2 + ... + (na) 2 } > n(na) 2 
> Z[a 2 + (2a) 2 + ... + (n~ la) 2 } ; 
lastly, Prop. 2 gives limits for the sum of n terras of the 
series ax + x 2 , a. 2 x + (2x) 2 , a. 3x + (3 x) 2 ,..., in the form of 
inequalities of ratios, thus : 
n {a. nx + (nx) 2 } : {a . rx + (rx) 2 } 
> (a + nx) : (%a + ^nx) 
I > n{a . nx + (nx) 2 } [a. rx + (rx) 2 ]. 
Prop. 3 proves that, if QQ' be a chord of a parabola bisected 
at V by the diameter P V, then, if P V be of constant length, 
the areas of the triangle PQ(/ and of the segment PQQ' are 
also constant, whatever be the direction of QQ'to prove it 
Archimedes assumes a proposition ‘ proved in the conics ’ and 
by no means easy, namely that, if QD be perpendicular to PV, 
and if p, p a be the parameters corresponding to the ordinates 
parallel to QQ' and the principal ordinates respectively, then 
QV 2 :QD 2 =p:p a . 
Props. 4-G deal with the area of an ellipse, which is, in the