Bookbinding - one form of hinge between the covers and the spine of a book.
A French groove is an indentation at the spine edge of the cover of a book. It is created by laying a spacer of some sort between the turn-up of the rounded and backed spine and the spine edge of the board before attaching the cover to the book block. The spacer can either be a metal rod (such as a knitting needle of the required diameter), or a raised metal half-cylinder on a specially made pressing board. (Guess which amateur bookbinders use!)
closed
French
groove
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| book block
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cover /
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| book block
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French grooves were invented, astonishingly enough, by bookbinders in France at the end of the 18th Century, at the same time as the hollow back. Before then, covers were attached to books using tight joints. French grooves provide a thin, flexible strip to act as a hinge between the cover and the book, and make the application of endpapers much easier. Along with hollow backs, they are unfashionable with fine bookbinders and indispensible to mass-market publishers.
Other types of grooved cover have been invented, such as the Harrison groove and the American groove, but these terms refer purely to construction methods. The final effect is still commonly called a French groove. | 
