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Page 5
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.
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