This book is A History of Dancing from the Earliest Ages to Our Own Times, 2nd Edition, by Gaston Vuillier, 1898. 446 p., published by D. Appleton & Co., NY.

It's in fair shape - the spine cover is loose on one side, spine is cracking, though page signatures are still intact.  One plate at the front is loose, other plates and pages are all good.

I can make out the following signatures (a few I can't) from members of the Ballet Russe de Monte Carlo:

Ruth Page

Herbert Bliss

Maria Tallchief

Paul Strauss

Alexandra Danilova

Shirley Haynes

Fernando Alonso

Alisia Alonso

Nora White 

Mary Ellen Moylan 

Igor Youskevitch  

Vicente Nebrada  

Nathalie Krassovska 

Igor Schwezoff 

Frederic Franklin 

Anton Dolin 

Nicholas Magallanes 

Nikita Talin

This book is from the library of Edith James Long (my grandmother), of the Edith James School of the Dance in Dallas Texas.  The dancers have noted a couple of dates - 1947 and 1966 at the latest.  I believe the occasion for the signing was a reunion of members of the ballet company with Edith James in 1966.  Members of the Ballet Russe came and went from its second incarnation in 1938, perhaps its most well-known name from the recent period being Yvonne Craig.

Edith James initially danced with the Denishawn Company as they went on a dance tour of Asia in 1925-26.  Returning to the US, she formed her own Dance School, where eventually Alexandra Danilova and George Balanchine taught workshops.  Thus the connection to the Ballet Russe.

Questions are welcome - I'll answer any I can.