AN IMPROVED PLANIMETER
Year: 1855

page_text: LETTERS PATENT to Rudolph Bodmer, of 2, Thavies Inn, in the City
of London, for the Invention of "AN IMPROVED PLANIMETER."-A commu-
nication from Jacob Amsler, Professor of Mathematics, of Schaffhausen,
in Switzerland.
Scaled the 22nd February 1856, and dated the 6th December 1855.
PROVISIONAL SPECIFICATION left by the said Rudolph Bodmer at
the Office of the Commissioners of Patents, with his Petition, ou the
6th December 1855.
I, RUDOLPH BODMER, of 2, Thavies Inn, in the City of London, do hereby
declare the nature of the said Invention of "AN IMPROVED PLANIMETER," (being
a communication from Jacob Amsler, Professor of Mathematics, of Schaff-
hausen, in Switzerland,) to be as follows:-
My improved "planimeter," which I call "polar planimeter," consists of
two bars or limbs, A and B of rectangular section, the latter of which is at one
end provided with an adjustable tracing point C, whilst the opposite end passes
through and is longitudinally moveable in a bush D. This bush D is
furnished with two brackets, which carry a horizontal spindle holding a
narrow cylinder E of about inch diameter at one end, and being furnished
at the other end with a worm gearing into a worm wheel attached to a short
perpendicular spindle, which carries a division plate, and is intended to
indicate the number of revolutions of the cylinder E (the circumference of

page_text: which is divided into 100 parts) during the operation of the instrument. The
length of the tracing point C must be so adjusted that when the limb B is
placed upon a level surface with the cylinder E downwards, it stands in a
perfectly horizontal position. The limb A also is at one end provided with an
adjustable pin F, whilst at the opposite end it hinges on a short upright
spindle G, which is held between the under side of the bush D and a corre-
sponding bracket, at a short distance from the end of the bush D in the
direction of the point C. On its upper face the limb B has straight transverse
lines, and the letters a, b, c, d, e, & f, engraved upon it. The distance from a
(which corresponds with the centre of spindle G) to the centre of the tracing
point C represents the height of a rectangle whose base is equal to the
circumference of the cylinder E, and whose area is one-tenth of a square foot,
6 C, c C, d C, e C, f C, being respectively the heights of rectangles whose
base is the circumference of the cylinder E, and whose areas are two thou-
sandths of an acre, ten square inches, five hundredths of a square foot, one
thousandth of an acre, and five square inches. These areas may be for
the sake of convenience engraved at or near the lines a, b, c, d, e, and f,
respectively, as 0-1 sq. f.; 2 ac. 1:1000; 10 sq. in.; 0.05 f; 1 ac. and 5 sq.
inch. The vertical sides of the limb B have lines engraved upon them
corresponding with those on the upper face, and are marked with numbers
which are determined in the following manner:-The limb A is swung
back on its centre until the centre of the pin F falls within a vertical
plane extending from the outer face of the cylinder E. In that case the
distance from C to F represents the radius of a circle whose area expressed
In square feet and multiplied by one-tenth produces the number indicated
at a. Had the limb B been moved back in the bush D until the line b
or the line c instead of the line a had intersected the centre of the spindle
G, then the distance from C to F would have represented the radius of a
circle whose area would have to be expressed in thousandths of an acre,
and to be multiplied with two or ten respectively in order to obtain the
number marked at 6 or c. Now, when this planimeter is intended to be used
for the measurement of the area of a figure drawn on paper, for instance, it is
necessary that the points C and F and the cylinder E should touch the paper,
and the instrument must be placed in such a manner that the outlines of the
figure can be followed with the tracing point C whilst the point F remains
stationary. After having pressed the point F slightly against the paper, the
point C is first placed upon any given point of the outline, whereupon the
numbers indicated on the division plate and on the nonius connected with
the cylinder E are read off and noted. The point C is then made to follow the

page_text: outline of the figure from left to right until it returns to its starting point, and
the numbers on the division plate and nonius are again noted. The difference
between the former and the latter reading is (when the point F lies without
the figure) the area of the figure, the unit being a rectangle whose base is
equal to the circumference of the cylinder E, and whose height is the distance
from the point C to the centre of the spindle G. If, on the contrary, the point
Flies within the figure whose area is to be measured, then to the difference
between the two readings must be added the area of a circle whose radius is
ascertained as above described. In order that it may not be necessary every
time to measure the distance from G to C, the limb B must, previously to
commencing the operation, be so placed that any one of the lines marked
a, b, c, &c., passes through the centre of the spindle G.
SPECIFICATION in pursuance of the conditions of the Letters Patent, filed
by the said Rudolph Bodmer in the Great Seal Patent Office on the
5th June 1856. .
TO ALL TO WHOM THESE PRESENTS SHALL COME, I, RUDOLPH
BODMER, of 2, Thavies Inn, in the City of London, send greeting.
WHEREAS Her most Excellent Majesty Queen Victoria, by Her Letters
Patent, bearing date the Sixth day of December, in the year of our Lord
One thousand eight hundred and fifty-five, in the nineteenth year of Her
reign, did, for Herself, Her heirs and successors, give and grant unto me, the
said Rudolph Bodmer, Her special licence, that 1, the said Rudolph Bodmer, my.
executors, administrators, and assigns, or such others as I, the said Rudolph
Bodmer, 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, use, exercise, and
vend, within the United Kingdom of Great Britain and Ireland, the Channel
Islands, and Isle of Man, an Invention for "AN IMPROVED PLANIMETER,"
(being a communication from Jacob Amsler, Professor of Mathematics, of
Schaffhausen, in Switzerland,) upon the condition (ainongst others) that I,
the said Rudolph Bodmer, my executors or administrators, by an instrument
in writing under my, or their, or one of their hands and seals, should parti-
cularly 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 imme-
diately after the date of the said Letters Patent.

page_text: NOW KNOW YE, that I, the said Rudolph Bodmer, do hereby declare the
nature of the said Invention, and in what manner the same is to be per-
formed, to be particularly described and ascertained in and by the following
statement (that is to say):-
This improved "planimeter," to be called polar planimeter," which, as its
name implies, is intended to measure the area of figures of every description
coming within its compass, distinguishes itself from other instruments of a
similar kind by its great compactness and simplicity, and by the facility with
which the area or superficies measured by means of it can be ascertained. In
order that the construction and application of the improved planimeter may be
clearly understood, I shall proceed to describe the same, reference being had
to the annexed Sheet of Drawings and to the letters and figures marked
thereon.
Fig. 1 is a ground plan; and Fig. 2, a side elevation of the improved
planimeter. One end of the bar or limb 1 passes through and is longitudinally
moveable in the bush 2, whilst the opposite end of it is furnished with a bush
3 containing an adjustable tracing point 4. From the bush 2 downwards
project the arms 5 & 6, betweer which the spindle 7 is held, by means of the
adjusting screws 8 & 9 (see Fig. 2), one end of the spindle 7 carrying the
drum or cylinder 10, whose surface is divided into 100 parts or degrees, whilst
the opposite end holds a worin 11, which gears into the worm wheel 12. The
worm wheel 12, bush 13, and division plate 14, which are or may be made of
one piece, are guided by the vertical spindle 15, which is screwed into the
bracket 16, the latter being attached to the arm 5 of the bush 2, and acting
at the same time as index to the division plate 14. Another bar or limb 17
is supported at one end by the short upright spindle 18, which is secured
between the under side of the bush 2 and a spur 19 projecting from the arm 5,
whilst the other end is furnished with a bush 20, wherein a steel pin 21 is
inserted. The division plate 14, the circumference of which is divided into
ten parts (see Fig. 1), is intended to indicate the number of revolutions of the
drum or cylinder 10 whilst the instrument is in operation; and it will be
perceived, that in order to obtain the required degree of accuracy in the
reading of the divisions marked on the drum 10, a nonius 22 is attached to
the arm 6 of the bush 2.
On the upper face of the bar or limb 1, and at the
points marked a, b, c, d, e, & f, (see Fig. 1,) straight transverse lines are en-
graved, and near to these in succession the marks 0.1 sq. f.; 2 ac. 1:1000;
10 sq. in.; 0.05 f.; 1 ac. 1.1000; 5 sq. in. And the distances from the centre
of the tracing point 4 are determined in the following manner :-The distance
4a represents the height of a rectangle whose base is equal to the outer

page_text: circumference of the flange 10¹ of the drum 10, and whose area is one-tenth
of a square foot. In like manner the distances 4 6, 4 c, 4 d, 4 e, and 4 f, are
respectively the heights of rectangles whose bases are equal to the outer
circumference of the flange 10 of the drum 10, and whose areas are 2 acres,
(on the scale of 1/1000); 10 square inches; 0.05 square foot; 1 acre, (on the
scale of 1/1000); and 5 square inches.
In that case
The figures 18.838, 19.120, and 20.252, marked on the side of the arm
or limb 1 (see Fig. 2), and the use of which will be hereafter explained, are
thus determined:-After having adjusted the bar or limb 1 in such a manner
in its bush 2 that the transverse line marked a (see Fig. 1) intersects the
centre line of the spindle 18, the instrument is placed in such a position that
the pin 21 falls within the extension of a vertical plane in which lies the
outer circumference of the flange 10¹ of the division drum 10.
the distance 21-4 represents the radius of a circle whose area expressed in
square feet and multiplied by 0·1 produces the figure 18.838 engraved at a.
Had the bar or limb 1 been adjusted in its bush so as to cause the line 6 or
the line c to intersect the axis of the spindle 18, then the distance 21-4 would
have represented the radius of a circle whose area would have to be expressed
in acres to the scale of 1/1000, and in square inches, and multiplied
two
or ten, in order to produce the figures 19.120 and 20.252 marked at b
and c.
Now, when this planimeter is intended to be used for the purpose of
measuring the area of a figure drawn on paper for instance, it is necessary
that the points 21 & 4 and the flange 10¹ of the division drum 10 should
simultaneously touch the paper, as shewn in Fig. 2; and the instrument must
be placed in such a manner that the outlines of the figure can be followed
with the tracing point 4 whilst the point 21 remains stationary. After having
pressed the pin 21 slightly against the paper, the tracing point 4 is first placed
upen any given part of the outline of the figure to be measured, whereupon
the numbers indicated on the division plate 14, the degrees on the division
drum 10, and the fraction marked by the nonius 22, are read off-and noted.
The tracing point 4 is then made to follow the outline of the figure from-left
to right until it returns to its starting point, and the numbers on the division
plate 14, drum 10, and nonius 22 are again noted. When the point 21 lies
without the figure to be measured, the difference between the former and the
latter reading is the area of the figure, the unit being a rectangle whose base
is equal to the outer circumference of the flange 10', and whose height is the
distance from the tracing point 4 to the centre of the spindle 18. If, on
the other hand, the point 21 lies within the figure to be measured, then to the

page_text: difference between the two readings must be added the area of a circle whose
radius is found in the manner already described. In order to avoid the
necessity of measuring at every operation the distance from the point 4 to the
centre of the spindle 18, any one of the lines a, b, c, &c., which may corre-
spond with the unit which is to serve as the measure of the area of the figure,
is to be placed at the point of intersection with the axis of the spindle 18,
a proceeding which offers at the same time this advantage, that the amounts to
be added, in cases where the point 21 lies within the figure to be measured, can
be read off without any trouble.
Supposing, for instance, that on the instrument having been placed in the
proper position for measuring the area of a given figure, the line c intersecting
the axis of the spindle 18, the division plate 14 marked 3, and the drum 10;
90-5, (90 degrees being read off direct from the drum, and the fraction 0.5
from the nonius,) the reading would be written thus 3.905. And if, after
following the outline of the figure in the manner described, the division plate
and drum were to mark 5.763 (i. e., the division plate indicated 5, and the
drum and nonius 76.3 degrees), then, in the case of the point 21 being with-
out the figure, 3.905 would have to be deducted from 5-763, leaving a differ.
ence of 1.858, which being multiplied by 10, would give the area of the figure
as 18.58 square inches. But, in the case of the point 21 being within the
figure, to the second reading, viz., 5.763, would have to be added 20.252,
as marked on the side of the bar or limb 1 corresponding with the line c.
Deducting from the sum 26-015 so obtained 3.905, the difference 22.110.
multiplied by 10=221.1 would be the area of the figure in square inches.
Had the line in the foregoing cases been placed on the point of intersection
with the axis of the spindle 18, that is, the line corresponding with 5 sq. in.,
then the differences would have had to be multiplied by 5 instead of by 10 in
order to obtain the area in square inches. In like manner if the marks a
(0.1 sq. f.) or 6 (0.05 sq. f.) had been placed at the point of intersection, the
differences of the readings would have had to be multiplied by 0-1 and 0.05
respectively. And if the plan of a field or piece of ground on the scale of
1/1000 had been submitted to the operation of the instrument, the difference
between the readings would have had to be multiplied by 2 if the line
b had been placed at the point of intersection. When the figure is drawn
upon a reduced scale, the difference of the reading must be multiplied by
the square of the ratio of reduction, for instance, by 10=100, when the
Drawing represents the figure upon the scale of one tenth of the natural
size only.
If, in the case of the piece of ground to be measured in acres, the scale,

page_text: instead of being in the proportion of 1 to 1000, were to be in the proportion of
1 to 5000, then the difference between the two readings would have to be
multiplied by ()=25; and in the case of the reduction having been made
upon the scale of 1 to 500, the multiplier would have to be (1) 2=1..
Attention has to be drawn to two sources of error in the reading of the
numbers on the division plate. If the index were, for instance, to point nearly
to the line of intersection between the divisions marked 3 and 4 on the
division plate 14, then the number 3 would have to be read off, supposing
that the drum 10 marked 100 degrees nearly; but the number 4 would have
to be taken if the drum 10 indicated 0 degree, or nearly so. The division
plate 14 may during the operation of the instrument perform one or even two
revolutions; it is therefore evident that if it should finally mark 7, the reading
must be either 17 or 27 according to the number of revolutions. Should the
division plate mark 6 and the drum and nonius 0° degree 7/10 or 4 degrees'
7/10, the reading would have to be 6.007 & 6.047, the surface of the drum 10
being divided into 100 degrees, as already observed.
Having now described the improved planimeter and the manner in which it
is to be used, I wish to observe that various instruments have heretofore been
made for measuring the areas of figures, but most of them are exceedingly
complicated, difficult to mariage, and very heavy and expensive. Those of the
most recent and approved construction consist of a foot plate provided with
parallel rails upon which moves a carriage carrying a disk of glass or brass of
from 3. to 6. ins. diameter, which is moveable about a vertical axis. The
carriage further supports a horizontal ruler of from 12 to 16 inches in length
guided by four rollers in a direction perpendicular to the rails upon which the
carriage moves. Along the ruler a band or string extends embracing a roller
or pulley, which is placed upon the same axis as the horizontal disk, and oue
end of this ruler is furnished with a tracing point reaching down to the
foundation upon which the foot plate stands. Upon the upper surface of the
horizontal disk rests the flange of a vertical roller or pulley, whose axis.is
parallel to the rails, and which turns upon centres in a frame in connection
with the foot plate. Other planimeters have a horizontal conus instead of the
disk. These instruments, however, do not conveniently admit of the measure-
ment of figures of a greater length then from 5 to 8 inches, nor of a greater
width than from 3 to 6 inches.
Now, by means of the polar planimeter, any figure can be measured whose
dimensions do not exceed those of a circle whose radius is equal to the sum of
the lengths 21-18 and 4-18 (see Fig. 1); and the limbs 1 & 17 can be made
of any desirable length. The instrument represented in the accompanying

page_text: Sheet of Drawings is capable of measuring a circle of two feet in diameter.
Moreover any pentograph can be converted into a planimeter of the improved
construction by the application of an indicator drum like that marked 10 and
10' in the accompanying Sheet of Drawings; all that is required for such con-
version being that the axis of the drum shall be parallel to some ruler not in
immediate connection with the fixed point of the pentograph.
What I claim as the Invention intended to be secured to me by the before-
mentioned Letters Patent is,-
Firstly, the constructing planimeters having neither a foot plate nor a
carriage, band or string, guide or supporting rollers, horizontal disk or conc.
Secondly, the mode of guiding the indicator drum, (marked 10 and 10' in
the accompanying Sheet of Drawings,) upon the plane upon which the figure
to be measured is drawn, in such a manner that the motion of the indicator
drum about its axis is the means of immediately indicating the area of the
figure whose outline the tracing point (marked 4 in the accompanying Sheet
of Drawings) follows; the mode of guiding the indicator drum being such
that the latter occupies in reference to a moveable straight line (the longitu-
dinal axis of the bar or limb 1) a parallel and invariable position, at the same
time that one extreme point of this line (the point of intersection with the
prolongation of the axis of the spindle marked 18 in the accompanying Sheet
of Drawings) describes a given curve (which in the instrument under
consideration is a circle, but which may be exchanged for any other curve),
whilst the other extreme point (the tracing point marked 4 in the accom-
panying Sheet of Drawings) traces the outline of the figure whose area is to
be measured.
And, lastly, I claim the mode of adjusting a part of the planimeter (the bar
or limb marked 1 in the accompanying Sheet of Drawings) so that the
planimeter shall indicate the area of any given figure in the measure of any
country, and upon any convenient scale without the necessity of changing
any part of the instrument.
In witness whereof, I, the said Rudolph Bodmer,. have hereunto set
my hand and seal, this Fifth day of June, in the year of our Lord
One thousand eight hundred and fifty-six.
RUDOLPH BODMER. (L.S.)