Henry S. H. (Henry Selby Hele) Shaw.

Mechanical integrators, including the various forms of planimeters online

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Prof. HENRY S. H. S.",

Reprinted from the Proceedings of the Institution of
Civil Engineers.

/ ^47 ^



23 MuBBAY AND 27 Wabebn ^''"" ■"

mn 19C8


, Mechanical aids to mathematical compu-
tation have always deservedly been regarded
with interest.

Aside from the labor-saving quality which
most of them possess, they have a value aris-
ing from the fact that they represent thoughts
of more or less complexity expressed in mech-

They are of many kinds, and serve widely
different purposes. The reader will find in
this essay descriptions of many that are use-
ful directly or indirectly to engineers.


Mechanical Integrators.

All measurements are made in terms
of some fixed unit. The method may
consist of a simple comparison of the
unit with the quantity to be mensured,
but wiien this cannot conveniently be
done, some indirect means must be em-
ployed. Indirect measurements may be
made by measuring some physical effect,
the magnitude of which is known to be
a function of the quantity to be meas-
ured ; as, for instance, when the length
of a rod or wire is estimated by its
weight. Where, however, the unit in
terms of which the measurement has
to be made, is what is known as a
derived unit, the indirect method gen-
erally consists in measuring in terms
of the simple units from w^hich the


former is derived, and performing,
with the results, the necessary calcula-
tion. An example of this latter method
is given in obtaining the contents of an
area, by taking its length and breadth
and multiiDlying them together, instead
of adopting the tedious process of ascer-
taining, by direct comparison, bow many
times the unit of area would be con-
tained within it.

Now, such calculations, even when of
so simple a kind as mere multiplication,
often become very inconvenient, and a
large number of instruments have been
designed for performing them by me-
chanical means. Such instruments may
be divided into two classes, one in which
the final result of conditions which vary
in an arbitrary manner is found, such as
the contents of a surface or the work of
a motor, requiring a process of multi-
plication or addition; the other in w^hich
the relation or ratio, at any instant, of
two such quantities is given, such as
space and time in the case of velocity,
requiring at each instant a process of

division. The object of the present pa-
per is to deal with the theory, design
and practical applications to engineering
problems of the former class alone. It may-
be briefly stated that very little has yet
been piractically done in the use of the
latter. Quite recently, Professor A. W.
Harlacher, of Prague, has published an
account of the instruments and methods
of Harlacher, Henneberg and Smreker,
for gauging the velocity of a river cur-
rent, the principle of which is the same
as that independently adopted by the
author and others in this country.

The conditions or data above referred
to, from which the required result has to
be calculated by instrumental means, are
obtained in two ways:

(1) By intermittent or separate obser-
vations and measurements.

(2) By the continuous motion of a
machine in connection with self-record-
ing apparatus.

The former is the case in measuring
an area of country, taking dimensions of
a river or embankment section, or ob-


taming the forces exerted at different
times by a machine or body in motion.
The latter is generally given in the form
of a graphic record, an important exam-
ple of which is the diagram of energy
or work taken from a prime mover. In
both cases the result, whatever it be,
whether boundary, area, volume, work,
etc., can be found by calculation, but
only with an approximation to the truth,
depending upon the extent of the calcu-
lation. The reason of this is that the
data of calculation, which are taken di-
rectly in the first case, or selected from
the graphic record in the second, only
represent actual conditions more or less
closely, according as the number of data
so taken is greater or less, and the great-
er the number the greater is the labor
of determining the result. The instru-
ments discussed in this jDaper perform
such work mechanically, with the great
advantages of rapidity of operation, ac-
curacy of results, and without requiring
mental effort on the part of the manipu-
lator; and all this, moreover, to a great

extent independently of the complexity
of the calculation required. All the re-
sults the measurement of which will be
considered, can be measured graphically.
If the observations have been made sep-
arately, tliey can be plotted, whether in
the form of a diagr im of energy, or on
the plan or elevation of an area or sec-
tion, and the boundary can be filled in
with a tolerably close approximation to
accuracy. In the other case the graphic
record is, or may be, directly given. The
subject, as far as the theory of the cal-
culation goes, can therefore be studied
with reference to such diagrams without
the necessity of considering in the first
case how they were obtained, and it will
be convenient to do this, and afterwards
to examine separately various examples
of their application. Such diagrams may
be drawn upon any kind of surface, and
an instrument for dealing with measure-
ments upon that of a sphere will be here-
after described. A plane surface may,
however, be employed upon which to re-
present all cases of any practical import-


ance, and the question thus arises, What
are the measurements of the nature un-
der consideration which are required, and
which can be obtained from either a reg-
ular or an irregular plane of figure?

Such measurements are of three
kinds :

(1) The length of its perimeter or

(2) The area of its superficial contents.

(3) Its relation to some point, line, or
other figure on the surface, e. g.^ its mo-
ment of area or moment of inertia about
a given line.

All these three kinds of quantities can
be ascertained by successive operations
of addition. The first requires the ad-
dition of elements of length, the second
may be obtained by adding up successive
elements in the form of strips of area,
and the third by adding products ob-
tained by multiplying such strips by
some quantity, the magnitude of which
depends upon the position of the other
point, line, or figure in question. Tak^
ing the general case of an irregular fig-


ure, it is evident that absolute accuracy
can only be obtained when this opera-
tion becomes that of integration or sum-
ming up of an infinite series of indefin-
itely small quantities. Instruments for
performing this operation are therefore
called '• mechanical integrators.'' In all
such instruments the rolling action of
two surfaces in frictional contact is em-
ployed, for this, as will be hereafter
seen, enables the conditions of motion
to be continuously varied in a way which
could not be effected by mere trains of
wheel-work, such as form the mechanism
of some kinds of calculating machines.
This fact necessitates something more
than a mere discussion of the mathe-
matical principles upon which the calcu-
lations are performed, for though the
action of an integrator may be absolute-
ly correct as far as its theory of the per-
formance of the calculation is concerned,
yet there is always some instrumental
error depending upon the rolling, and
also, as will be seen, of the slipping of
the two surfaces in frictional contact.


This error may be exceedingly small. Init
it is a matter of great importance to
ascertain its exact amount, and the sub-
ject will therefore be investigated at
length, under the heading "Limits of
Accuracy of Integrators," where an ac-
count will be also given of the experi-
mental results of Professor Lorber, of
Leoben; Dr. William Tinter, of Vienna,
and Dr. A. Amsler, of Schaflfhausen. In
this investigation it will be showu that
when integrators are examined upon the
mechanical principles of action, they are
all found to belong to one of two clasBes.

(1) In which the surfaces in question
slip over each other.

(2) In which only pure rolling motion
of the surfaces is assumed to take place.

The significance of this mode of clussi-
fication is that it not only leads to a cl&ir
understanding of the nature of the re-
sults to be expected from any particular
instrument, and teaches the best method
of manipulating it, with regard to its po-
sition relatively to the figure to be meas.
ured, but it also brings out prominently


tlie meolianiral principle uptui wliicb tlie
inventor has relied sometimes, as it
NVouM appear, unoonsciouHly, for the ac-
ruracv of tlie r«->»tiltH »\jnMt« d to be ob-

It may be here remarked that the wime
priiJcipU', by which an intej^'rator is em-
j)h)yo(l to determine a result from an au-
to^'ni]>!iic record, may Ih) apphed directly
to «»btain a continuous result from the
machine or bo at the extremity
of the arm is made to pass round the
boundary of the figure, the disk wall be
turned through a distance proportional


to the travel along OX, while at any in-
stant the roller {m) is at the same dis-
tance from the center of the disk as the
pointer is from OX.

If 2/^z=CB=: mean height of element A. x' =
= width of element AB.
Then, by the reasoning already given?
the reading of the roller which the
pointer passes over the upper boundary
of the element AB, is

71^ — y^Ax,
and the final reading of the roller is
N=area of the figure A DDE.
Hansen, in 1850, still further improved
this instrument, and, in conjunction with
Ausfeld, introduced a different method of
reading the result, and of carrying the
frame, this instrument being known as
the Han sen- Ausfeld planimeter. Various
other instruments of the same kind were
shown in the Great Exhibition of 1851,
but in all, the motion of the arm carrying
the pointer was " linear ; '' that is, the
motion, which must be possible in every
direction, is obtained by compounding
two rectilinear movements, at right angles



to each other. Such instruments are
therefore called " Imear planimeters."

Many different forms of linear planim-
eters have been suggested, but the only
modification of the disk and roller which
it will be worth while to notice is the
cone and roller.

Let MM', Fig. 4, be the cone cor-
responding to the disk M, and rolling
on the edge of its two bases in a direction
parallel to OX. Let the roller ni always
be in contact with a circle on the cone,
whose center B' is at a distance CB'
from the apex C of the cone, such that

CB' = SB = ?/=mean ordinate of ele-
ment SB.
where the element AB is being at that
instant integrated. Adopting the same
notation as before, when the cone has
rolled over the surface through a distance
Ace, then, whatever be the angle of its
apex, the distance rolled through by the
roller m is




As might have been anticipated, the
expression is the same as was obtained in
the case of the disk, the hitter being a
special case of the cone when the vertical
angle is 180°.

Thus the cone may be employed in-
stead of the disk, and such an instrument
was invented by Mr. E. Sang, who, in
1852, published a description of it, ac-
cording to which the action was extremely
accurate, but it does not appear to have
come into very extensive use.

No more instruments of the kind will
be described, since they have given place
to those in which the arm carrying the
pointer turns about a center or pole, and
which are, therefore, called "polar plan-

In the year 1856, Professor Amsler-
Laflbn invented and brought before the
world the now well-known polar planim-
eter bearing his name, and, since then,
no less than twelve thousand four hun-
dred of these instruments have been
made and stnt out from his works at
Schaffhausen. According to authorities,


which Professor Lorber quotes, Professor
Miller, of LeobeD, invented independently
a planimeter of this kind in the same
year (1856), which, being made by Starke,
of Vienna, is known as the Miller-Starke
planimeter. Previous to this, in 1854,
Decher, of Augsberg, as well as Bounia-
kovsky, of St. Petersburg (1855), had

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