The late mathematician and musical amateur, W. S. B. Woolhouse, no less than Fétis, contributed greatly to a full understanding of the essential properties of a bow on the part of those whose office it is to produce the actual instrument. Woolhouse laid great stress on a point overlooked by many other students of the subject, the same being that the success of a bow depends quite as much on its purity as a vibrating body as does the violin.

Unless the bow is so adjusted in its weight and proportions that it vibrates with absolute uniformity throughout its entire length it is useless to an artist.

Bows are "false" frequently in the same way that strings are. Inequalities of finish, imperceptible to our ordinary senses, will render a perfect "staccato" from end to end impossible, just as it is impossible to obtain true fifths in every part of a violin's compass if one of the strings be slightly wanting in absolute cylindricity. I speak specially of "staccato," as that form of bowing suffers perhaps more than any other from faulty bows; but any form of bowing that calls for special dexterity will betray the inefficiency of a bow.

It is of great interest to compare the calculations of Woolhouse with those of Fétis, and I will here quote the results obtained by the former.

"If measurements be taken in inches, and parts of an inch, and h denote the distance of any part of the bow from the head, the diameter of the stick in that locality, supposing the bow to be round, may be readily calculated from the following formula:—

Diameter = .2 [log.(h + 7.25) - 9.8100]

"From this formula the numbers given in the last column of the following table were calculated."

Distance from Head of Bow in Inches.
Violin Viola Violoncello Diameter
in parts of
an inch.
0
2
4
6
9
13
18
23
0
1½
3
5
8
11½
15
19
23

0
1
3
5½
9
12
16
20
24		 .210
.230
.247
.262
.280
.300
.318
.333
.348
.360
.370

These measurements, of course, only extend to the commencement of the cylindrical portion.

Woolhouse made a small gauge of ivory, based on the above measurements, which proved of great practical value in examining bows. The measurements he obtained by the above calculation apply to wood of medium density. He says, "For close and dense wood the dimensions should be somewhat diminished, or, what amounts practically to the same thing, the distance from the head should, for dense wood, be increased by half an inch, or an inch, as the case may be, before applying the gauge." He then gives a table of inclusive weights of violin, viola and violoncello bows.