See if this helps. Look at the circle as a clock. The Clock has 12 numbers and music has 12 note titles (choosing the proper enharmonic key in which they are written). 7 natural and 5 accidental titles to determine the proper sequence with the numbers on the clock.
C Major scale uses all 7 Natural notes in it's make up, so it's the strongest scale to start in clockwise position. # 12 is top of the clock, so Root C starts the sequence. The sequence of Natural notes in C Major (C-D-E-F-G-A-B) all have something in common, : All carry the title of Whole tone, except one. F gets a Half tone Title, while D-E-G-A-B, get Whole tone status. They also share Natural notes as their 4th interval (D=G, E=A, G=C, A=D, B=E)
Root C's 4th is also a natural note, C=F... F is the only Natural that has an accidental 4th interval. F=Bb. The Odd Duck. F is next to C, but has less in common with C's other neighbor, G. So, G gets position #1 clockwise (and F gets position number 11 counter clockwise). Next clockwise # 2 position closest to G is D. Next to D is A (# 3), Next to A is E (# 4) , Next to E is B (#5).
Going in sequence counter clockwise the Next position to F is Bb (number 10),
Next to Bb is Eb (number 9), Next to Eb is Ab (number 8), Next to Ab is Db(number 7).
This brings us to the number 6 position of the Clock and the bottom of the circle where clockwise and counter clockwise collide into an accident....... Both directions can title the key to maintain the sequence of their accidentals being understood and can even overlap to claim the # 5 or number 7 position.
Going counter clockwise from number 7 (Db position), next position to Db is Gb (number 6) and then counter clockwise Overtaking # 5 position (B) from Gb (number 6) would be re-titled Cb (number 5).
Going clockwise from # 5 position (B), next to B is F# (#6), then overtaking position number 7 (Db position) clockwise from F# (#6) would be re-titled C# (#7).
Notice that counting the positions away from key of C indicates the number of accidentals the next key in sequence contains; Clockwise, G = 1# to C# = 7#'s. Counter clockwise F = 1b to Cb = 7b's.
This theory pertains to the sequence of notes in which music is written.
Notice the difference when following the sequence of notes by trying to write the Major scale using D# and Eb (Enharmonic pitch, D#/Eb)
D#-E#-F##-G-A-B-C#. Using the sequence of notes doesn't work with the proper interval pattern.
The correct notes from D# would be; D#-F-G-G#-A#-C-D-D#, but written in this manner would be confusing and sporatic to the Stave.
Now Eb
Eb-F-G-Ab-Bb-C-D-Eb; all notes follow sequence with the interval pattern and can be written with understanding.