Thomas Brown did not publish any description of his instrument, but his
son, John Brown, published in 1661 a small book,[12] in which he says
(preface) that he had done “as Mr. Oughtred with Gunter’s Rule, to a
sliding and circular form; and as my father Thomas Brown into a Serpentine
form; or as Mr. Windgate in his Rule of Proportion.” He says also that
“this brief touch of the Serpentine-line I made bold to assert, to see if I
could draw out a performance of that promise, that hath been so long
unperformed by the promisers thereof.” Accordingly in Chapter XX he gives a
description of the serpentine line, “contrived in five (or rather 15)
turn.” Whether this description, printed in 1661, exactly fits the
instrument as it was developed in 1632, we have no means of knowing. John
Brown says:
1. First next the center is two circles divided one into 60, the other
into 100 parts, for the reducing of minutes to 100 parts, and the
contrary.
2. You have in seven turnes two inpricks, and five in divisions, the
first Radius of the sines (or Tangents being neer the matter, alike to
the first three degrees,) ending at 5 degrees and 44 minutes.
3. Thirdly, you have in 5 turns the lines of numbers, sines, Tangents, in
three margents in divisions, and the line of versed sines in pricks,
under the line of Tangents, according to Mr. Gunter’s cross-staff: the
sines and Tangents beginning at 5 degrees, and 44 minutes where the other
ended, and proceeding to 90 in the sines, and 45 in the Tangents. And the
line of numbers beginning at 10, and proceeding to 100, being one entire
Radius, and graduated into as many divisions as the largeness of the
instrument will admit, being 10 to 10 50 into 50 parts, and from 50 to
100 into 20 parts in one unit of increase, but the Tangents are divided
into single minutes from the beginning to the end, both in the first,
second and third Radiusses, and the sines into minutes; also from 30
minutes to 40 degrees, and from 40 to 60, into every two minutes, and
from 60 to 80 in every 5th minute, and from 80 to 85 every 10th, and the
rest as many as can be well discovered.
The versed sines are set after the manner of Mr. Gunter’s Cross-staff,
and divided into every 10th minutes beginning at 0, and proceeding to 156
going backwards under the line of Tangents.
4. Fourthly, beyond the Tangent of 45 in one single line, for one Turn is
the secants to 51 degrees, being nothing else but the sines reitterated
beyond 90.
5. Fifthly, you have the line of Tangents beyond 45, in 5 turnes to 85
degrees, whereby all trouble of backward working is avoided.
6. Sixthly, you have in one circle the 180 degrees of a Semicircle, and
also a line of natural sines, for finding of differences in sines, for
finding hour and Azimuth.
7. Seventhly, next the verge or outermost edge is a line of equal parts
to get the Logarithm of any number, or the Logarithm sine and Tangent of
any ark or angle to four figures besides the carracteristick.
8. Eightly and lastly, in the space place between the ending of the
middle five turnes, and one half of the circle are three prickt lines
fitted for reduction. The uppermost being for shillings, pence and
farthings. The next for pounds, and ounces, and quarters of small
Averdupoies weight. The last for pounds, shillings and pence, and to be
used thus: If you would reduce 16s. 3d. 2q. to a decimal fraction, lay
the hair or edge of one of the legs of the index on 16. 3½ in the line of
1. s. d. and the hair shall cut on the equal parts 81 16; and the
contrary, if you have a decimal fraction, and would reduce it to a proper
fraction, the like may you do for shillings, and pence, and pounds, and
ounces.
The uses of the lines follow.
As to the use of these lines, I shall in this place say but little, and
that for two reasons. First, because this instrument is so contrived,
that the use is sooner learned then any other, I speak as to the manner,
and way of using it, because by means of first second and third
radiusses, in sines and Tangents, the work is always right on, one way or
other, according to the Canon whatsoever it be, in any book that treats
of the Logarithms, as Gunter, Wells, Oughtred, Norwood, or others, as in
Oughtred from page 64 to 107.
Secondly, and more especially, because the more accurate, and large
handling thereof is more then promised, if not already performed by more
abler pens, and a large manuscript thereof by my Sires meanes, provided
many years ago, though to this day not extant in print; so for his sake I
claiming my interest therein, make bold to present you with these few
lines, in order to the use of them: And first note,
1. Which soever of the two legs is set to the first term in the question,
that I call the first leg always, and the other being set to the second
term, I call the second leg . . .
The exact nature of the contrivance with the “two legs” is not described,
but it was probably a flat pair of compasses, attached to the metallic
surface on which the serpentine line was drawn. In that case the instrument
was a slide rule, rather than a form of Gunter’s line. In his publication
of 1661, as also in later publications,[13] John Brown devoted more space
to Gunter’s scales, requiring the use of a separate pair of compasses, than
to slide rules.