Practical problems are different in various respects 
from purely mathematical problems, yet the principal
motives and procedures of the solution are essentially the
same. Practical engineering problems usually involve
mathematical problems. We will say a few words about
the differences, analogies, and connections between these
two sorts of problems.
1.   An impressive practical problem is the construction
of a dam across a river. We need no special knowledge
to understand this problem. In almost prehistoric times,
long before our modern age of scientific theories, men
built dams of some sort in the he valley of the Nile, and in
other parts of the world, where the crops depended on
irrigation.
     Let us visualise the problem of constructing an important
modern dam. 
    What is the unknown?  Many unknowns are involved
in a problem of this kind: the exact location of the dam
its geometric shape and dimensions, the materials used
in its construction, and so on.
      What is the condition We cannot answer this question
in one short sentence because there are many conditions.
In so large a project it is necessary to satisfy many important
 economic needs and to hurt other needs as little
as possible. The dam should provide electric power, sup
ply water for irrigation or the use of certain communities
and also help to control floods.
     On the other hand, it should disturb as little as possible 
navigation, or economically important fish-life, or beautiful scenery;
and so forth. And, of course, it should cost as little as possible
and be constructed as quickly as possible.
     What are the data?  The multitude of desirable data is
tremendous. We need topographical data concerning the
vicinity of the river and its tributaries; geological data
important for the solidity of foundations, possible leakage,
and available materials of construction; meteorological
data about annual precipitation and the height of
floods;economic data concerning the value of ground
which will be flooded, cost of materials and labour; and so on.
     Our example shows that unknowns, data, and condititions
are more complex and less sharply defined in a 
practical problem than in a mathematical problem.
     2. In order to solve a problem, we need a certain
amount of previously acquired knowledge. The modern
engineer has a highly specialised body of knowledge at
his disposal, a scientific theory of the strength of mate
materials, his own experience, and the mass of engineering
experience stored in special technical literature. We can-
not avail ourselves of such special knowledge here but
we may try to imagine what it was in the mind of an 
ancient Egyptian dam-builder.
      He has seen, of course, various other, perhaps smaller,
dams: banks of earth or masonry holding back the water.
He has seen the flood, laden with all slots of debris,
pressing against the bank. He might have helped to re-
pair the cracks and the erosion left by the flood. He
might have seen a dam break, giving way under the
impact of flood.  He certainly heard stories about
dams withstanding the test of centuries or causing catastrophe
by an unexpected break. His mind may have
pictured the pressure of the river against the surface of
the dam and the strain and stress in  its interior.
      Yet the Egyptian dam-builder had no precise, quanta-
tative, scientific concepts of fluid pressure or of strain and 
stress in a solid body. Such concepts form an essential
part of the intellectual equipment of a modern engineer.
Yet the latter also uses much knowledge which has not
yet reached a precise, scientific level;
what he knows about erosion by flowing water, the transportation 
of silt, the plasticity and other not quite clearly circum
scribed properties of certain materials, is knowledge of 
a rather empirical character.
     Our example shows that the knowledge needed the
concepts used are more complex and less sharply defined
in practical problems than in mathematical problems.
     3. Unknowns, data, conditions, concepts, necessary
preliminary knowledge, everything is more complex 
and less sharp in practical problems than in purely mathematical
 problems.  This is an important difference
perhaps the main difference, and it certainly implies further
differences; yet the fundamental motives and procedures
of the solution appear to be the same for both sorts of
problems.
     There is a widespread opinion that practical problems
connered more experience than mathematical problems. This 
may be so. Yet, very likely, the difference lies in the
nature of the knowledge needed and not in our attitude
toward the problem. In solving a problem of one or the
other kind, we have to rely on our experience with similar
 problems and we often ask the questions:Have you 
seen the same problem in a slightly different form? Do
you know a related problem?
     In solving a mathematical problem, we start from very
clear concepts which are fairly well ordered in our mind.
In solving a practical problem, we are often obliged to
start from rather hazy ideas; then, the clarification of the
concepts may become an important part of the problem.
Thus, medical science is in a better position to check
infectious diseases today than it was in the times before
Pasteur when the notion of infection itself was rather
hazy.   Have you taken into account all essential notions
involved in the problem?  This is a good question for all
sorts of problems but its use varies widely with the nature of 
the intervening notions.
     In a perfectly stated mathematical problem all the data
and all clause of the condition are essential and must be
taken into account. In practical problems we have a multitude
ti of data and conditions; we take into account as 
many as we can but we are obliged to neglect some. Take
the case of the definer of the large dam. He considers the
public interest and important economic interests but he
is bound to disregard certain petty claims and grievances.
The data of his problem are, strictly speaking, inexhaustible-
. For instance, he would like to know a little
more about the geologic nature of the ground on which
the foundations must be laid, but eventfully he must
stop collecting geologic data, although a certain margin
of uncertainty unavoidably remains.
     Did you use all the data?  Did you use the whole
condition? We cannot miss these questions when we deal
with purely mathematical problems. In practical problems,
however we should put these questions in a modified-
 form: Did you use all the data which could con-
tribute apreciably to the solution ? Did you use all the 
conditions which could influence appreciably the solu-
tion? We take stock of the available relavant informa-
tion, we collect more information if necessary, but
eventually we must stop collecting we must draw the 
line somewhere, we cannot help neglecting something
"If you will sail without danger, you must never put to
sea?" Quite often, there is a great surplus of which
have no appreciable influence on the final form of the 
solution.
4.     Ther desiners fo the ancient Egyptian dams had to 
rely on the common-sense intepretation of their experie-
ence, they had nothing else to rely on. The modern
engineer cannot rely on comon sense alone, especially 
if his project is of a new and daring design; he has to cal
culate the resitance of thhe projected dam, foresee quan-
titatively the strain and the stress in its interior. For this
purpose, he has to apply the thoery of elasticity( which
applies fairly well to constructions in concrete). To
aplly this theory, he needs a good deal of mathematics;
the practical engineering problem leads to a mathemati-
cal problem.
      This  mathematical problem is too technical to be dis-
cussed here; all we can say about it is a general remark.
In setting up and in solving mathematical problems de-
rived from pracitcal problems, we usually content
ourselves with an approximation. We are bound to neglect
some minor data and conditions of the practical
problem. Therefor it s reasonable to allow some slight
inaccuracy in the computations especially when we can
gain in simplicity what we lose in accuracy.
5. Much could be said about approximations that
would deserve general interest. We cannot suppose, 
however, any specialised mathematical knowledge and
therefor we restrict ourselves to just one intuitive and 
instructive example.
     The drawing of geographic maps is an important practical
problem. Devising a map, we often assume that the
earth is a sphere. Now this is true only as an approximate
assumption and not the exact truth. The surface of the 
earth is not at all a mathematically defined surface and
we definitely know that the earth is flattened at the poles
Assuming, however, that the earth is a sphere,we may
draw a map of it much more easily. We gain much in
simplicity and do not lose a great deal in accuracy. In 
fact let us imagine a big ball that has exactly the shape
of the earth and that has a diameter of 25 feet at its
equator. The distance between the poles of such a ball
is less than 25 feet because the earth is flattened, but only 
about one inch less. Thus, the sphere yields a good practical
approximation.