This probably was a good book in 1907, but too much has changed for it to be valuable now. The author's repeated references to square pianos, "international" vs. "concert" tuning frequencies, and tuning to C-517.3 are comically antiquated.
I appreciate a mathematical approach to tuning, but the discussions in chapters 8-14 are a clumsy attempt to explain concepts that someone with a decent mathematics and music background can easily calculate -- that the ratio between adjacent notes is the 12th root of 2; and that chord, interval, and beat relationships can be derived from that ratio. A modern calculator makes much of this book completely pointless.
The author's math sequences are cumbersome, and his conclusions are sometimes wrong. His beat frequency calculations are not always correct. The beat frequency of his C-128 to G-191.78 combination should be .44 Hz., not .66 as he calculates. A simpler and more correct way to calculate the beat frequency of a 5th interval is to subtract the 2nd harmonic (2 X 191.78) of the higher string from the 3rd harmonic of the lower (3 X 128).
The discussion on where/how to place mutes is helpful, as are the points about taking proper steps to ensure the validity of piano hardware before tuning. Also, the method of setting temperament by 5th and octave steps is useful.
In summary, if you have enough math and music experience to see through the dated material, the cumbersome derivations, and the false conclusions, this book gives a few helpful tips. I would not recommend it to most tuning amateurs, however.