As I fear the title of my paper to our Society to-night contains two
misstatements of fact in its three words, I must commence by
correcting it. In the first place, the instrument to which I propose
to draw your attention to-night is, in the narrow sense of the words,
neither an integrator nor new. The name "integrator" has been
especially applied to a class of instruments which measure off on a
scale attached to them the magnitude of an area, arc, or other
quantity. Such instruments do not, as a rule, represent their results
graphically, and we may take, as characteristic examples of them,
Amsler's planimeter and some of the sphere integrating machines.
An integrator which draws an absolute picture of the sum or integral
is better termed an "integraph." The distinction is an important and
valuable one, for while the integraph theoretically can do all the
work of the integrator, the latter gives us in niggardly fashion one
narrow answer, _et præterea nil_. The superiority of the integraph
over the integrator cannot be better pointed out than by a concrete
example. The integrator could determine by one process, the bending
moment, from the shear curve, at any one chosen point of a beam; the
integraph would, by an equally simple single process, gives us the
bending moment at all points of the beam.
In the language of the mathematician, the integrator gives only that
miserly result, a definite integral, but the integraph yields an
indefinite integral, a picture of the result at all times or all
points--a much greater boon in most mechanical and physical
investigations. Members of our Society as students of University
College have probably become acquainted with a process termed "drawing
the sum curve from the primitive curve." Many have probably found this
process somewhat wearisome; but this is not an unmixed evil, as the
irksomeness of any manual process has more than once led to the
invention of a valuable machine by the would-be idler. Thus our innate
desire to take things easy is a real incentive to progress. It was
some such desire as this on my part which led me, three years ago, to
inquire whether a practical instrument had not been, or could not be,
constructed to draw sum curves. Such an instrument is an integraph,
and the one I have to describe to you to-night is the outcome of that
inquiry. It is something better than my title, for it is an integraph,
and not an integrator.
[Illustration: A NEW INTEGRATOR]
Before I turn to its claims to be considered new, I must first remind
you of the importance of an instrument of this kind to the
draughtsman. I put aside its purely mechanical applications, where it
has been, or can be, attached to the indicators of steam engines, to
dynamometers, dynamos, and a variety of other instruments where
mechanical integration is of value. These lie entirely outside my
field, and I propose only to refer to a few of the possible services
of the integrator when used by hand, and not attached to a machine.
The simple finding of areas we may omit, as the planimeter will do
that equally well. But of purely graphical processes which the
integraph will undertake for us, I may mention the discovery of
centroids, of moments of inertia (or second moments), of a scale of
logarithms, of the real roots of cubic equations, and of equations of
higher order (with, however, increasing labor). Further, the
calculation of the cost of cutting and embanking for railways by the
method of Bruckner & Culmann, the solution of a very considerable
number of rather complex differential equations, various problems in
the storage of water, and a great variety of statistical questions may
all be completely dealt with, or very much simplified by aid of the
integraph.
In graphical statics proper the integraph draws successively the
curves of shear, bending moment slope, and deflection for simple
beams; it does the like service for continuous beams, after certain
analytical or graphical calculations have first been made; it can
further lighten greatly the graphical work in the treatment of masonry
arches and of metal ribs. In graphical hydrostatics it finds centers
of pressure and gives a complete solution for the shear and bending
moment, curves in ships, besides curves for their stability. In
graphical dynamics the applications of the integraph seem still more
numerous. It enables us to pass from curves of acceleration to curves
of speed, and from curves of speed to curves of position. Applied to
the curve of energy of either a particle or the index point of a rigid
body, it enables us by the aid of easy auxiliary processes to
ascertain speeds and curves of action. In a slightly altered form,
that of "inverse summation," we can pass from curves of action to
curves of position, and deal with a great range of resisted motions,
the analysis of which still puzzles the pure mathematician; the
variations of motion in flywheels, connecting rods, and innumerable
other parts of mechanism, may all be calculated with much greater ease
by the aid of an integraph. Shortly, it is the fundamental instrument
of graphic dynamics.
It would be needless to further multiply the instances of its
application; the questions we have rather to ask are: Can a practical
instrument be made which will serve all these purposes? Has such an
instrument been already put upon the market? If I have to answer these
questions in the negative, it is rather a doubtful negative, for the
instrument I have to show you to-night goes so far, and suggests so
many modifications and possibilities, which would take it so much
further, that it is very close to bringing the practical solution to
the problem.
Let me here lay down the conditions which seem essential to a
practical integraph. These are, I think, the following:
1. The price must be such that it is within the reach of the ordinary
draughtsman's pocket. The Amsler's planimeter at £2 10s. or £3 may be
said to satisfy this first condition. The price for the first complex
integraph designed by Coradi was £24 to £30. The modified form in
which I show it to-night is estimated to cost retail £14. Till an
equally efficient instrument can be produced for £5 I shall not
consider the price practical. If the error of its reading be not
sensibly greater than that of a planimeter, it is certainly worth
double the money.
2. The instrument must not be liable to get out of order by fair
handling and a reasonable amount of wear and tear. I cannot speak at
present with certainty as to how far our integraph satisfies this
condition; it is rather too complex to quite win my confidence in this
respect.
3. It must be capable of being used on the ordinary drawing board, and
of having a fairly wide range on it, i.e., it must not be limited to
working where the primitive is at one part only of the board.
This condition takes out of every day practical drawing use the
integraph invented by Professors James and Sir William Thomson, in
which the sum curve is drawn on a revolving cylinder. It is essential
that the sum curve should be drawn on the board not far from the
primitive, and that this sum curve can be summed once or twice again
without difficulty. The time involved in drawing the four sum curves,
for example, required in passing from the load curve to the deflection
curve of a simple beam, if these curves were drawn on different pieces
of paper and had to be shifted on and off cylinders, would probably be
as long as the ordinary graphical processes. Coradi's integraph works
on an ordinary drawing board, but since there are nearly 10 inches
between the guide point and tracer, the sum curve is thrown 10 inches
behind the primitive in each integration. Thus a double summation
requires say 26 inches of board, and it is impossible to integrate
thrice without reproducing the primitive. The fact that the primitive
and sum curve are not plotted off on the same base is also troublesome
for comparison, and involves scaling of a new base for each summation.
I have endeavored to obviate this by always drawing the second sum
curve on a thin piece of paper pinned to the board, which can then be
moved back to the position of the first primitive. But this shifting,
of course, involves additional labor, and is also a source of error.
I should like to see the trace and guide chariots on the same line of
rails, one below the other, were this possible without producing the
bad effect of a skew, pull or push.
4. The practical integraph must not have a greater maximum error than
2 per cent. The mathematical calculations, which are correct to five
or six places of decimals, are only a source of danger to the
practical calculator of stresses and strains. They tend to disguise
the important fact that he cannot possibly know the properties of the
material within 2 per cent. error, and therefore there is not only a
waste of time, but a false feeling of accuracy engendered by human and
mechanical calculation which is over-refined for technical purposes.
For comparative purposes I have measured the areas of circles of 1
inch, 2 inches, and 3 inches radius, the guide being taken round the
circumference by means of a "control lineal," first with an ordinary
Amsler's planimeter and then with the integraph. I have obtained the
following results:
---------+------------+-----------+-----------------------------------
| | | By integraph.
Radius | | By |--------+--------+--------+--------
of | Calculated |Planimeter.| | Upper | | Upper
circle. | areas. | |Middle. | end. |Middle. | end.
| | |p=2 in. |p=2 in. |p=4 in. |p=4 in.
---------+------------+-----------+--------+--------+--------+--------
in. | | | | | |
1 | 3.14159 | 3.140 | 3.140 | 3.138 | 3.120 | 3.120
| | | | | |
2 | 12.56636 | 12.55 | 12.36* | 12.546 | 12.568 | 12.552
| | | | | |
3 | 28.27431 | 28.24 | ...... | ...... | 28.280 | 28.288
---------+------------+-----------+--------+--------+--------+--------
* Cross bar had to be moved during tracing.
From this it follows that the error of the planimeter is less than 0.1
per cent. and that of the integraph about 0.5 per cent. Obviously we
could make this error much less if we excluded small areas measured
with large polar distances, or such polar distances that the cross bar
must be shifted. Excluding such cases, we see that the accuracy of the
integraph scarcely falls behind that of the planimeter and is quite
efficient for practical purposes. It must be borne in mind that the
above measurements were made with the "control lineal," an arrangement
which carries the guide round a circle of the exact test area. In most
cases the curve has to be followed by hand, and the error will be
greater--greater probably for the integraph than for the planimeter,
as the former is distinctly hard to guide well.
I think, then, we should be safe in saying that the error of the
integraph is not likely to be greater and is probably less than 2 per
cent., so that in this respect the instrument may be considered a
practical one.
5. A further condition for a good integraph is that it should have a
wide range of polar distances, and that it should be easily set at
those distances.
One of the conditions I gave to the maker of the instrument was that
it should be able to take all polar distances from one to ten
half-inches. This condition he can scarcely be said to have fulfilled.
With polar distances of 1/2 inch and 1 inch, the machine works
unsatisfactorily, which indeed might have been foreseen from the
construction of its sliding bars. It works best from 2.5 inches to 5
inches, and this is the range to which I think we ought to confine the
present type of instrument. As the last conditions I may note that: