We give here an ordered list of all types of particle accelerators and exhibit how each type evolves conceptually and/or technologically from the preceding. This is in contrast to the usual “history of particle accelerators” in which unrelated accelerator types are listed in the chronological order. It is hoped that this discussion and understanding of the rationale and logic in the evolution of one accelerator type to the next will help to educe future inventions. We give here an ordered list of all types of particle accelerators and exhibit how each each evolves conceptually and / or technologically from the preceding. This is in contrast to the usual “history of particle accelerators ” in which unrelated accelerator types are listed in the chronological order. It is hoped that this discussion and understanding of the rationale and logic in the evolution of one accelerator type to the next will help to educe future inventions.