ARCHIMEDES 
touches it at one point only. For, if possible, let the tangent 
at P touch the spiral at another point Q. Then, if we bisect 
the angle POQ by OL meeting PQ in L and the spiral in P, 
0P + 0Q = 20P by the property of the spiral. But by 
the property of the triangle (assumed, but easily proved) 
OP + OQ > 2 OL, so that OL < OP, and some point of PQ 
lies within the spiral. Hence PQ cuts the spiral, which is 
contrary to the hypothesis. 
Props. 16, 17 prove that the angle made by the tangent 
at a point with the radius vector to that point is obtuse on the 
‘ forward ’ side, and acute on the ‘ backward ’ side, of the radius 
vector. 
Props. 18-20 give the fundamental proposition about the 
tangent, that is to say, they give the length of the suhtangent 
at any point P (the distance between 0 and the point of inter 
section of the tangent with the perpendicular from 0 to OP). 
Archimedes always deals first with the first turn and then 
with any subsequent turn, and with each complete turn before 
parts or points of any particular turn. Thus he deals with 
tangents in this order, (1) the tangent at A the end of the first 
turn, (2) the tangent at the end of the second and any subse 
quent turn, (3) the tangent at any intermediate point of the 
first or any subsequent turn. We will take as illustrative 
the case of the tangent at any intermediate point P of the first 
turn (Prop. 20). 
If OA be the initial line, P any point on the first turn, FT 
the tangent at P and OT perpendicular to OP, then it is to be 
proved that, if ASP be the circle through P with centre 0, 
meeting PT in S, then 
(subtangent OT) = (arc ASP). 
I. If possible, let OT be greater than the arc ASP. 
Measure off OU such that OU > arc ASP but < OT. 
Then the ratio PO : OU is greater than the ratio PO : OT, 
i. e. greater than the ratio of \ PS to the perpendicular from 0 
on PS. 
Therefore (Prop. 7) we can draw a straight line OQF meeting 
TP produced in F' and the circle in Q, such that 
FQ : PQ = PO : 0 U.