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To find the fraction of the book James read each night, we start by dividing the total fraction of the book he read, which is \( \frac{6}{7} \), by the number of nights he read, which is 8.
The amount he read each night can be calculated as:
\[
\text{Fraction read each night} = \frac{\frac{6}{7}}{8}
\]
We can express 8 as a fraction:
\[
\text{Fraction read each night} = \frac{6}{7} \div 8 = \frac{6}{7} \times \frac{1}{8}
\]
Now we multiply the fractions:
\[
\text{Fraction read each night} = \frac{6 \times 1}{7 \times 8} = \frac{6}{56}
\]
Next, we simplify \( \frac{6}{56} \):
\[
\frac{6}{56} = \frac{3}{28}
\]
Thus, the fraction of the book that James read each night is:
\[
\frac{3}{28}
\]
Therefore, the answer is:
\[
\boxed{\frac{3}{28}}
\]