mary problems of uncertain inference will have reached its complete
solution. If not, there must remain
some further puzzles to unravel."
Uncertain inference refers to the
mathematics of probability, in
which great strides have been made in the last two centuries in approximations of the unknown.
Professor E. B. Wilson, Harvard
mathematician, stated that the
problem put by Dr. Fisher was, in
his opinion, similar to some of the
important economic problems with
which the world is confronted to-
For many years Dr. Fisher has
been a guiding spirit in the scientific work of the Rothamated agricultural experiment station in England, and is now galton professor
He has contributed to the theory
of statistics, notably to the theory
of small samples and in the development of the notion of efficiency
whereby the maximum of information can be culled from the data.
He has designed improved layouts for agricultural experimentation and has contributed to the gen-
etical theory of natural selection.
Nature of the Kasner Discovery
Dr. Kasner's discovery is an outgrowth of his research on the horn
angle, one of the age-old mathematical mysteries, which had been
classed by the ancient Greeks and
all mathematicians for the succeeding 2,500 years with such other
enigmas of geometry as the squaring of the circle and the trisection
of an angle. Solution of this horn
angle problem was announced by
Dr. Kasner Wore the National
Academy of Sciences at Charlottesville, Va., in November.
The horn angle is that formed
when two or more curved lines
branch out of a common stem, thus
forming the shape of a horn. The
ancient Greeks decided the horn
angle had zero value and therefore
could neither be measured nor bisected. Newton and his successors
decided, after trying, that the
Greeks were right.
Dr. Kasner, by applying a technique discovered about a hundred
years ago by the German mathe
matician, Gauss, found a new type
of measuring rod for the horn
angle. By applying this method he
discovered to his great surprise
that th emeasure of the parts, when
added together, did not add up to
the sum of the whole, but was generally much greater than the
whole.
There were some instances, he
reported, in which the parts, when
added up, were equal to the whole.
But in no instance were the parts
found to be less than the whole.
Modifies His "Measuring Rod"
This must be described as a new
type of geometry, belonging to non-
Archimedian geometry," Dr. Kasner said. In Archimedian geometry
all quantities are comparable, so
that, if two quantities are given,
in which one is greater than the
other, the smaller could be made
equal to or greater than the larger
by multiplying it enough times.
Thus, if one has two quantities,
one of which is a 100 times greater
than the other, the smaller can be
made equal to the greater by multiplying it by 100 or making definite
additions to it until the smaller
reaches the measure of the larger.
This is true with ordinary angles,
which are measured by degrees.
But in horn angles, which cannot
be measured in degrees, but must
be measured by the abstract measuring rod discovered by Dr. Kasner,
it is not possible, he found, to make
a small horn angle equal a larger
one by the process of multiplication or addition.
In fact, Dr. Kasner found, one
can multiply a small horn angle by
any suitable finite number without making it equal to the larger
horn angle.
Another paradoxical discovery
made by Dr. Kasner is that if a
horn angle is bisected each of the
halves will be the equal of the
Dr. Kasner also reported a modification of his original measuring
rod which he announced before the
National Academy of Sciences. The
measuring rod consists of "variables." Dr. Kasner employs four
such variables. Two are the estimated curvatures of the two curves
of the horn angle, while the other
two are the rates of variation in the two curvatures.