The property of the lever, from its fundamental principle, was
studied by him, beginning with the self- evident fact that "equal
bodies at the ends of the equal arms of a rod, supported on its
middle point, will balance each other"; or, what amounts to the
same thing stated in another way, a regular cylinder of uniform
matter will balance at its middle point. From this starting-point
he elaborated the subject on such clear and satisfactory
principles that they stand to-day practically unchanged and with
few additions. From all his studies and experiments he finally
formulated the principle that "bodies will be in equilibrio when
their distance from the fulcrum or point of support is inversely
as their weight." He is credited with having summed up his
estimate of the capabilities of the lever with the well-known
expression, "Give me a fulcrum on which to rest or a place on
which to stand, and I will move the earth."

But perhaps the feat of all others that most appealed to the
imagination of his contemporaries, and possibly also the one that
had the greatest bearing upon the position of Archimedes as a
scientific discoverer, was the one made familiar through the tale
of the crown of Hiero. This crown, so the story goes, was
supposed to be made of solid gold, but King Hiero for some reason
suspected the honesty of the jeweller, and desired to know if
Archimedes could devise a way of testing the question without
injuring the crown. Greek imagination seldom spoiled a story in
the telling, and in this case the tale was allowed to take on the
most picturesque of phases. The philosopher, we are assured,
pondered the problem for a long time without succeeding, but one
day as he stepped into a bath, his attention was attracted by the
overflow of water. A new train of ideas was started in his
ever-receptive brain. Wild with enthusiasm he sprang from the
bath, and, forgetting his robe, dashed along the streets of
Syracuse, shouting: "Eureka! Eureka!" (I have found it!) The
thought that had come into his mind was this: That any heavy
substance must have a bulk proportionate to its weight; that gold
and silver differ in weight, bulk for bulk, and that the way to
test the bulk of such an irregular object as a crown was to
immerse it in water. The experiment was made. A lump of pure gold
of the weight of the crown was immersed in a certain receptacle
filled with water, and the overflow noted. Then a lump of pure
silver of the same weight was similarly immersed; lastly the
crown itself was immersed, and of course--for the story must not
lack its dramatic sequel--was found bulkier than its weight of
pure gold. Thus the genius that could balk warriors and armies
could also foil the wiles of the silversmith.

Whatever the truth of this picturesque narrative, the fact
remains that some, such experiments as these must have paved the
way for perhaps the greatest of all the studies of
Archimedes--those that relate to the buoyancy of water. Leaving
the field of fable, we must now examine these with some
precision. Fortunately, the writings of Archimedes himself are
still extant, in which the results of his remarkable experiments
are related, so we may present the results in the words of the
discoverer.

Here they are: "First: The surface of every coherent liquid in a
state of rest is spherical, and the centre of the sphere
coincides with the centre of the earth. Second: A solid body
which, bulk for bulk, is of the same weight as a liquid, if
immersed in the liquid will sink so that the surface of the body
is even with the surface of the liquid, but will not sink deeper.
Third: Any solid body which is lighter, bulk for bulk, than a
liquid, if placed in the liquid will sink so deep as to displace
the mass of liquid equal in weight to another body. Fourth: If a
body which is lighter than a liquid is forcibly immersed in the
liquid, it will be pressed upward with a force corresponding to
the weight of a like volume of water, less the weight of the body
itself. Fifth: Solid bodies which, bulk for bulk, are heavier
than a liquid, when immersed in the liquid sink to the bottom,
but become in the liquid as much lighter as the weight of the
displaced water itself differs from the weight of the solid."
These propositions are not difficult to demonstrate, once they
are conceived, but their discovery, combined with the discovery
of the laws of statics already referred to, may justly be
considered as proving Archimedes the most inventive experimenter
of antiquity.

Curiously enough, the discovery which Archimedes himself is said
to have considered the most important of all his innovations is
one that seems much less striking. It is the answer to the
question, What is the relation in bulk between a sphere and its
circumscribing cylinder? Archimedes finds that the ratio is
simply two to three. We are not informed as to how he reached his
conclusion, but an obvious method would be to immerse a ball in a
cylindrical cup. The experiment is one which any one can make for
himself, with approximate accuracy, with the aid of a tumbler and
a solid rubber ball or a billiard-ball of just the right size.
Another geometrical problem which Archimedes solved was the
problem as to the size of a triangle which has equal area with a
circle; the answer being, a triangle having for its base the
circumference of the circle and for its altitude the radius.
Archimedes solved also the problem of the relation of the
diameter of the circle to its circumference; his answer being a
close approximation to the familiar 3.1416, which every tyro in
geometry will recall as the equivalent of pi.

Numerous other of the studies of Archimedes having reference to
conic sections, properties of curves and spirals, and the like,


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