$50 
Mr T. Tredgold on Steam-Boats, 
will give the degrees in the angle contained between the two 
paddles. From unity substract the natural cosine of this angle, 
and the depth of the paddles divided by the remainder will 
give the radius of the wheel. 
Thus, if the number of paddles be 8, and their depth 1 A feet ; 
then-—- = 45°, and its cosine is -7071, therefore — 
5 1 — 7071 
5*1$ feet, the radius of the wheel. 
Again, if the number of paddles be 7, and their depth 15 feet 
360 
as before, then — = 51° 26' ; and its cosine is '6234, conse- 
quently 
1*5 
•6$34 
= 4 feet. 
Both divisions are represented in Fig. 1., and it may be re- 
marked, that, when the depth of the paddles is fixed upon, the 
greater number of paddles should have the preference, because 
the first impression on the water is then less vertical. The dif- 
ference is easily seen, by comparing the angles at which the pad- 
dles A and a , Fig. 1. strike the water. It will also be observed, 
that the larger wheel must have less tendency to throw the wa- 
ter up behind at C. 
It is obvious, that, by enlarging the wheel, the obliquity of 
the action on entering the water may be reduced, but it also 
may be done by lessening the depth of the paddles, as will be 
evident from Figs. 3. and 4., where the angles are the same in 
both wheels ; hence it is useful to be able to find the depth, 
and if the number of the paddles and the radius of the wheel 
be given, the depth may be found by the following rule : 
Multiply the radius of the wheel by the difference between 
unity and the natural cosine of the angle contained between two 
paddles, and the product is the depth required. Suppose the 
radius is to be 4-5 feet, and that there are to be eight paddles, 
then 4*5 (l — *7071) = 1*318 feet for the depth of the paddles. 
I think eight paddles is as small a number as ought to be 
adopted, and where large wheels can be admitted, nine or ten 
might be used with advantage, but where many paddles are em- 
ployed, the wheels must necessarily be of large diameter, to keep 
them narrow. The advantages of wheels of large diameter con- 
sist in the favourable direction they strike the water, and also 
