A history of Greek mathematics - Wilbourhall.org

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ON CONOIDS AND SPHEROIDS 57
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 1 8 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—l)a}
(this is clear from S n
= ^n(n + I) a)
(Lemma)
(n + 1) (na) 2 +-a(a + 2a + 3a + ... + na)
= 3 {a 2 + (2a) 2 + (3a) 2 + ... + (no) 2 }
whence (Cor.)
(Lemma to Prop. 2)
3 {a 2 + (2a) 2 + (3a) 2 + ... +(na) 2 } > n(na) 2
> 3{a 2 + (2a) 2 + ... + (n-la) 2 } ;
lastly, Prop. 2 gives limits for the sum of n terms of the
series ax + x 2 , a.2x + (2x) 2 , a . 3 x + (3 x) 2 ,
. . , .
in the form of
inequalities of ratios, thus
n{a.nx + (nx) 2 2/'" 1 } {a.rx + (rx) 2 : }
> (a + nx) : (\a + \nx)
> n { a . nx + (nx) 2 } 2
W :
X { a .
rx + (rx) 2 }
Prop. 3 proves that, if QQ' be a chord of a parabola bisected
at Fby the diameter PV, then, if PV be of constant length,
the areas of the triangle PQQ' and of the segment PQQ f are
also constant, whatever be the direction of QQ' ; to prove it
Archimedes assumes a proposition proved in the conies ' ' and
by no means easy, namely that, if QD be perpendicular to PV,
and if p, pa be the parameters corresponding to the ordinates
parallel to QQ' and the principal ordinates respectively, then
Props. 4-6 deal with the area of an ellipse, which is, in the