I have a 1960s Peugeot bike with a cottered bottom bracket that I want to convert into square tapered. I was told that this could easily be done just by replacing the spindle with a square tapered one. I measured the dimensions of the cottered spindle and found that the center, or B in this image from Sheldon Brown

was 56mm. If I buy a spindle with the middle length of 52mm, im think it would not work because the space between the bottom bracket shell and the spindle thing (dont know what its called) is too far apart for the loose bearings.

How do I solve this problem?

Buy a new french threaded bottom bracket? (or maybe just new bearing cups?)

or will the 52mm spindle work with the bottom bracket cups I already have?