A NEW METHOD OF FINDING AND INDICATING THE MEASUREMENTS OF THE CART AND COSTVES OF THE ARCS OF CIRCLES OR OTHER PERIPHERIES."
Year: 1853

page_text: LETTERS PATENT to Henry Montague, Grover, of Hitcham Rectory,
in the County of Buckingham, Clerk, for the Invention of "A
NEW METHOD OF FINDING AND INDICATING THE MEASUREMENTS OF THE
CART AND COSTVES OF THE ARCS OF CIRCLES OR OTHER PERIPHERIES."
PROVISIONAL SPECIFICATION left by the said Henry Montague
Grover at the Office of the Commissioners of Patents, with his
Petition, on the 13th July 1858.
I, HENRY MONTAGUE GROVER, of Hitcham Rectory, Bucks, do
hereby declare the nature of the said Invention for "A NEW METHOD OF
FINDING AND INDICATING THE MEASUREMENTS OF THE SINES AND COSINES OF
THE ARCS OF CIRCLES OR OTHER PERIPHERIES" to be as follows:-
My Invention relates generally to the application by mechanical
means of the chords or subtense lines of one circle to measure the sines
or half chords of another circle of double the magnitude.
For example, describe an arc B, C, as at Fig. 1, and from A, the
centre draw the radii A, C. A, B, at right angles to each other. Take

page_text: half the radius of B, C, and at the points D, D, as centres describe
semicircles A, E, B, and A, E, C; draw the radius A, E, F, to arc B, C.
The chord or subtense A, E, of the arcs of circles A, E, C, and A, E, B,
will be equal to the sine and cosine of the arc B, F, or the arc C, F.
Again draw the radius A, K; the chord A, L, of the arc of
circle A, E, C, now now produced will be equal to the sine K, M, of
the arc K, B, and the chord A, I, of circle A, E, B, will be equal to
the cosine K, N, of the same are K, B. Any radius of the quadrant
upon the semicircles will give a similar result.
It is the mechanical application of this principle to sextants and other
instruments of observation and survey, or to the measurement of sines
& cosines of arcs for any other practical purpose, which constitutes
my Invention; and in order to exemplify its application it may be
reduced to practice as follows, or by other suitable means:
Fig. 2, represents three hands, the combined action of which find the
sinc and cosine of arcs within a quadrant, as before mentioned. A, is
the centre of motion, on which I place a hand R, which I call the
index of any length- as the radius of a quadrant. The other two
hands S, S, which I call pointers, have the centres of motion disposed,
one on either point D, D, distant from the centre A, half the length of
index R, and on lines at right angles to each other, as before mentioned.
These pointers are just half the length of the index R, and are con-
nected with it in such a manner that while the index R, describes a
quarter circle. the pointers S, S, each describe a semicircle, and all
move in the same direction. A radial line is produced on the index R,
which line is graduated from the centre A, to show the length of the
sines and cosines of ares within the quadrant of the larger circle by a
measure counted from the centre A, to the point or points indicated by
the said pointers. The sines and cosines are indicated simultaneously,
the sine by the one hand and the cosine by the other, and this upon
any radii within the quadrant to which the index may be turned.
Other arrangements of apparatus may be employed to obtain like
results, & of which the foregoing is in illustration.

page_text: SPECIFICATION in pursuance of the conditions of the Letters Patent,
filed by the said Henry Montague Grover in the Great Seal
Patent Office, on the 18th January 1854.
'
TO ALL TO WHOM THESE PRESENTS SHALL COME, I,
HENRY MONTAGUE GROVER, of Hitcham Rectory, in the County of
Buckingham, Clerk, send greeting.
WHEREAS Her most Excellent Majesty Queen Victoria, by Her
Letters Patent, bearing dato the Thirteenth day of July, in the year of
our Lord One thousand eight hundred and fifty-three, in the seventeenth
year of Her reign, did, for Herself, Her heirs and successors give and
grant unto me, the said Henry Montague Grover, Her special licence
that I, the said Henry Montague Grover, my executors, administrators,
and assigns, or such others as T, the said Henry Montague Grover,
my executors, administrators, and assigns, should at any time agree
with, and no others, from time to time and at all times thereafter
during the term therein expressed, should and lawfully might make,
usc, exercise, and vend, within the United Kingdom of Great Britain
and Ireland, the Channel, Islands, and Isle of Man, an Invention for
"A NEW METHOD OF FINDING AND INDICATING THE MEASUREMENTS OF THE
SINES AND COSINES OF THE ARCS OF CIRCLES AND OTHER PHERIPHERIES." upon
the condition (amongst others) that I, the said Henry Montague Grover,
by an instrument in writing under my hand and seal, should particularly
describe and ascertain the nature of the said Invention, and in what
manner the same was to be performed, and cause the same to be filed
in the Great Seal Patent Office within six calendar months next and
immediately after the date of the said Letters Patent.
NOW KNOW YE, that I, the said Henry Montague Grover, do
hereby declare the nature of my said Invention, and in what manner
the same is to be performed, to be particularly described and ascer-
tained in and by the following statement, reference being had to the
Sheet of Drawings hereunto annexed, and to the letters and figures
marked thereon (that is to say):
Invention consists generally in the mechanical application of the
chords or subtense lines of one circle for the purpose of measuring the
sines or half chords of another circle double the magnitude of the first.

page_text: For example, describe an arc B, C, as at Fig. 1, and from A, the
centre draw the radii A, C, A, B, at right angles to each other. Take
half the radius of B, C, and at the points D, D; as centres, describe
semicircles A, E, B, and A, E, C; draw the radius A, E, F, to are B, C.
The chord or subtense A, E, of the arcs of circles A, E, C, and A, E, B,
will be equal to the sine and cosine of the arc C, F, or the are B, F.
Again, draw the radius A, K, the chord A, L, of the arc of circle
A, E, C, now produced will be equal to the sine K, M, of the are K, B,
and the chord A, I, of the circle A, E, B, will be equal to the cosine
K, N, of the same arc K, B. Any other radius of the quadrant upon
the semicircles will give a similar result.
The mechanical application of this principle to sextants and other
instruments of survey and observation, or the measurement of sines and
cosines of arcs for any other practical purposes, constitutes the principal
feature of my Invention, and in order to exemplify its application it may
be reduced to practise as follows, or by any other suitable means.
There are various mechanical means of practicable application for
finding and measuring the sines and cosines of arcs according to my
Invention, and as instances thereof, affording good and effectual means
of determining the sines and cosines of arcs as above, I will proceed to
describe the two following instruments.
An instrument for measuring the sines and cosines of arcs may be
formed in the following manner :-It is represented in plan at Fig. 2,
and in section at Fig. 3, in the Drawing annexed. Upon a plate o, of
metal, wood, or other material, and of suitable shape and dimensions,
1 engrave or otherwise indicate the two sides of a square A, B, A,
C, and upon the interior sides or lines of the said figure I engrave
or indicate two semicircles, whose centres are placed one on each
radii or side of the square, having their diameters of any length,
but made equal to the radii of the quadrantal arc, as indicated,
subtending the right angle of the said right angled figure. And I
provide two rulers or measuring ferrules P: P: 1, 1, equal to the length
of the diameters of the said semicircles or quadrantal radii as afore-
said, which said rulers or ferrules are divided on their fiducial
clges forming radii to the quadrant into any suitable number of
.divisions for measuring the length of the chords or subtense lines of

page_text: the said semicircles in passing over their faces. I locate the said rulers
or incasuring ferrules when not in use at the sides, and in a line
with the two diameters of the said semicircles, but beyond them in
respect to the interior of the quadrant, on which they : are placed in
recesses s, s, t, t, made for that purpose, so that the divided edges of the
said rulers should both abut with the utmost exactitude upon the angle
of the right lined figure or terminating point where the diameters of the
two semicircles meet and intersect the point A, forming the centre of
the quadrant. The said rulers are mounted on A, as a fulcrum, on
which they move radially; this, therefore, forms their common centre;
the divided edges, consequently, at all positions form radii to the quad-
rant. I provide that the said rulers may be made to move from their
places in a circular direction horizontally over the faces of the said semi-
circles by any means which will effect the object. For instance, I form a
circular or quadrantal groove r, 1, about midway of the semicircle's
diameter, from one diameter to the other of the said semicircles, with a
sliding guide u, attached to the rulers and fitted to the said groove in a
suitable manner to run in the said groove, and carry the rulers in the
circular direction, by which contrivance, if properly executed, the
divided edges of the said rulers will always appear in their progress
longitudinally on the faces of the said semicircles in the position of
radii to the said quadrantal arc. According to this Invention the said
rulers in passing over the faces of the said semicircles will shew at any
given point the length of the chords or subtense lines of the said two
semicircles by the divided measures upon their edges, thereby disclosing
the corresponding length of the sines and cosines of the arcs of the
said quadrantal circumference, which are determined by the points of
intersection of the radius with the semicircles. The distance of
the one intersection from the point A, being equal to the sine, and
that of the other to the cosine, as before exemplified. Thus will be
disclosed the sines and cosinės of any ares found in connection with
the diameters of the semicircles of relative proportion, of any circum-
ference or horizon, described round their centre of motion.
Fig. 5, represents an elevation, and Fig. 4, a plan of another
arrangement of apparatus, which for most practical purposes will be

page_text: more available than the foregoing; it consists of three hands or arms
mounted on a suitable frame working in permanent and invariable
connection with each other. a, b, are plates suitably fitted together,
forming the frame in which the axes c, d, e, are supported and turn
freely, carrying the hands f, g, h. The hand f, is the index, which may
be of any suitable length; it noves about on the axes c, while the
point or apex describes the quadrant, the sines and cosines of whose arcs
it is required to determine.
The hands h, g, are traversing pointers which give the reading on the
index at any point within the quadrant. These pointers are just equal
to half the length of the index f, measuring from their several centres
of motion to the point. While the index hand f, describes the quad-
raut the pointers describe the semicircles A, E, B, A, E, C, as referred
to in the foregoing Figures; their motions being so controled that
uniformity of action takes place. For this purpose I place their
several centres in connection by means of finely cut toothed wheels i, k, l,
i, k, l; the wheels k, l, are mere intermediate wheels for the purpose of
transmitting and reversing the direction of motion; the wheels 1, 1, are
just half the diameter, and furnished with half the number of teeth of
the wheel i, so that the wheels 1, 1, perform exactly half a revolution,
while i, describes one fourth of a revolution; the several hands conse-
quently describe corresponding revolutions.
Thus it will be seen that the pointers g, h, have their centres of
motion on the points D, D, before given, and are distant from the
centre of motion A, exactly half the length of the index hand f.
The hands all move round in the same direction, the motion being
such that the apices of the pointers travel in coincidence with the
graduated line of the index throughout the whole motion necessary to
describe the semicircles A, E, B, A, E, C, shown at Fig. 1. The
index is graduated from the centre A, to a scale sufficiently minute,
and according to the purpose for which it may be required. By count-
ing from the centre A, to points indicated by the pointers h, and g,
they will severally indicate the measures of the sines and cosines of
the ares of the quadrant, determined by the position of the index
hand. The sines and cosines are thus exhibited simultaneously, the

page_text: sine by the distance of one hand or pointer from the centre A, and the
cosine by the distance of the other pointer from center A, and this.
upon any radii within the quadrant to which the index hand f, may
be turned, which will be apparent on following out the different posi-
tions of the hands represented by the dotted lines.
In applying this apparatus to instruments of observation the frame-
work a, b, is fixed to the stationary or other part forming the basis of
the instrument, while the axis c, at A, of the index hand f, is placed
in such connection with the moveable or observing part of the instru-
ment that it may receive a like and corresponding amount of rotation
to which the instrument may be subject. It is necessary to observe
great care in fitting and adjusting the several parts of this instrument.
as on that the accuracy of the readings will entirely depend, the prin-
ciple in itself being perfect.
The graduation of the scale or index will depend upon the purposes
to which the instrument is to be applied, the finer the divisions and the
greater the accuracy of construction the more perfect will be the
result, which, as before mentioned, is mere measure of length counted
from the centre A, to the roading of the hands at any given point to
which the index hand may be turned, such measures of length being
equal to the sine and cosine of the arcs of the quadrant, as determined
by the then position of the index line graduated on the hand f.
The object it is sought to attain is exact the measure of the sines
and cosines at any angle of the instrument without the trouble of
calculation and deduction from the observation made, which is effected
simultaneously therewith.
It may also serve as a check against the observation, if carried out in
the ordinary way.
Having described the nature of my Invention, and the manner in
which the same is to be performed, I would have it understood that
what I claim is, the method of measuring the sines and cosines of arcs
or other peripherics by any quadrant, semicircle, or circle, or instru-
ment formed in connection with such quadrant, semicircle, or circle, as
shall have a divided index or ruler moving on a centre over the face of
such quadrant, semicircle, or circle as before first described, for the

page_text: purpose of measuring the chords or subtense lines of such quadrant,
semicircle, or circle, as a means of finding the sines and cosines of an
exterior circle or periphery. I also claim herein secondly described
for a like purpose, or any mechanical equivalent therefore.
In witness whereof, I, the said Montague Grover, have here-
unto set my hand and seal, this Twelfth day of January, in
the year of our Lord One thousand eight hundred and fifty-
four.
HENRY MONTAGUE GROVER. (L.S.)