234 
POPULAR SCIENCE REVIEW. 
CDEFGABCDEFGABCDEFGABCDEFGABC 
7 6 5 4 3 2 1 
We get in the order marked by figures, 
Series IV. 
0, F, B, E, A, D, G, C. 
Now, on inspecting this series, it is seen that they are the 
same notes , reversed in order , which were got in Series I. 
This (Series IV.) is, however, like Series I., not quite complete 
at first. Its 6 last fifths need alteration. For in its 2nd fifth 
2 semitones occur, and we must therefore get another semi- 
tone to perfect that fifth. This is to be done by flattening its 
last note, B, one semitone, which cuts into the succeeding fifths 
and requires their last notes to be flattened also. The Series 
then becomes 
Series V. 
0, F, bB, bE, bA, bD, bG, bC, (or B). 
Here C flat being the same as B natural, we may take B 
instead of C flat ; and if we continue the fifths below that, we 
get the following complete series, ending at last on another C 
natural; after which the series of fifths would only repeat 
themselves. 
Series VI. 
C,F, bB, bE, bA, bD, bG, B, E, A, D, G, C. 
0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12. 
Thus, in either rising (Series III.) or falling (Series VI.) by 
fifths from C natural we come in each case again to a C natural, 
after 12 fifths . 
VIII. We are now in a position to apply this to practical 
results. Taking Series II., if I wish to know what keys require 
1 , 2, or more sharps to indicate them in their signatures, this 
Series II. gives what we want. 
For it is found on trial that 
The key of C has 0 sharps. 
„ ,, G has 1 sharp. 
„ „ D has 2 sharps. 
,, ,, A has 3 sharps. 
„ „ E has 4 sharps. 
„ „ B has 6 sharps. 
„ „ #F has 6 sharps. 
„ „ 8(J has 7 sharps. 
Each note therefore in Series III. needs 1 sharp more than 
the note preceding it. 
