This is the 10446th most frequent Russian word.
версией
The Russian word "версией" means "version" in English (instrumental or genitive singular form).
Here, 'версией' is used in the instrumental case to indicate the tool or method (a new version) of the presentation of the book.
Эта книга представлена новой версией.
This book is presented with a new version.
In this context, 'версией' is used in the instrumental case describing a characteristic of the method discussed (the old version).
Мы обсудили проблему старой версией метода.
We discussed the problem with the old version of the method.
'Версией' in this sentence is the instrumental case, indicating the context in which ideas are applied (towards the new version).