ON SPIRALS 
69 
B in F. Then 
Conoids and Spheroids involving the summation of the series 
l 2 + 2 2 q- 3 2 + ... + n 2 . Prop 11 proves another proposition in 
summation, namely that 
G.BT, 
(n— 1) {na) 2 : {a 2 + (2a) 2 + (3a) 2 + ... + {n — l)«) 2 } 
> {na) 2 : {na.a +|{na — a) 2 } 
> {n— 1) {na) 2 : {(2a) 2 + (3a) 2 +... + {na) 2 ]. 
?. 
vcns assumed in 
The same proposition is also true if the terms of the series 
are a 2 , {a + b) 2 , {a + 2b) 2 ... {a + n — lb) 2 , and it is assumed in 
the more general form in Props. 25, 26. 
Archimedes now introduces his Definitions, of the spiral 
itself, the origin, the initial line, the first distance {= the 
radius vector at the end of one revolution), the second distance 
( = the equal length added to the radius vector during the 
second complete revolution), and so on; the first area (the area 
bounded by the spiral described in the first revolution and 
the ‘ first distance’), the second area (that bounded by the spiral 
geometers fall 
i, plane problem 
es, or generally 
;ase e.g. (1) with 
>s of Apollonius 
xedes assumes in 
’ character with 
it calling in the 
heorem given by 
inference of the 
il to the straight 
meet the tangent 
described in the second revolution and the ‘ second distance'), 
and so on; the first circle (the circle with the ‘first distance’ 
as radius), the second circle (the circle with radius equal to the 
sum of the ‘first’ and ‘second distances’, or twice the first 
distance), and so on. 
Props. 12, 14, 15 give the fundamental property of the 
spiral connecting the length of the radius vector with the angle 
through which the initial line has revolved from its original 
position, and corresponding to the equation in polar coordinates 
r — a 6. As Archimedes does not speak of angles greater 
than 7r, or 2 77, he has, in the case of points on any turn after 
the first, to use multiples of the circumference 
es, that he only 
does not assume 
i solution of the 
onius wrote two 
by Archimedes’s 
of such problems 
of a circle as well as arcs of it. He uses the p' . 
‘first circle’ for this purpose. Thus, if P, Q 
are two points on the first turn, / ( \I/ ^ 
OP :0Q = (arc AKP'): (arc AKQ'); 
if P, Q are points on the nth turn of the 
spiral, and c is the circumference of the first circle, 
;o Prop. 2 of On 
OP : OQ = {(n— l)c + arc AKP'} : {{n— l)c + arc AKQ'}. 
Prop. 13 proves that, if a straight line touches the spiral, it