substituting this into the second equation gives W5. It’s easy to check
that these numbers solve the problem; a garden with a length of 10 yards
and a width of 5 yards has an area of 50 square yards, and the length ex-
ceeds the width by 5 yards.
However, there is a second solution to the above quadratic equation;
L5. Substituting this into the second equation gives W10, and
this pair of numbers gives a satisfactory mathematical solution to the pair
of equations. Search as you might, though, you’re not going to find a gar-
den with a width of negative 10 yards—because width is a quantity that is
inherently positive.
The high-school student knows what to do in this case: discard the solu-
tion W10 and L 5, precisely because it is meaningless in the con-
text of the problem. If such an equation were to occur in physics, the
physicist would not be quite so quick to cast aside the apparent meaning-
less solution. Rather, he or she might wonder if there was some hidden
underlying meaning to the apparently “meaningless” solution that was
yet to be revealed, as there is a rich history of interesting physics underly-
ing apparently meaningless solutions.
The Gap in the Table
The dictionary definition of mathematics is usually similar to the one
I found in my ancient Funk & Wagnalls—the study of quantity, form,
magnitude, and arrangement. When an arrangement manifests itself so
that, in part, it explains phenomena in the real world, exploration is often
undertaken to see whether undiscovered phenomena correspond to miss-
ing parts of the arrangement. A classic such case is the discovery of the
periodic table of the elements.
In the nineteenth century, the chemists were attempting to impose order
and structure to the apparently bewildering array of the chemical ele-
ments. Dmitry Mendeleyev, a Russian chemist, decided to try to organize
the known elements into a pattern. To do so, he first arranged these ele-
ments in increasing order of atomic weight, the same physical property
that had attracted the attention of John Dalton when he devised the atomic
theory. He then imposed another level of order by grouping the elements
according to secondary properties such as metallicity and chemical reac-
tivity—the ease with which elements combined with other elements.
The result of Mendeleyev’s deliberations was the periodic table of the
elements, a tabular arrangement of the elements in both rows and col-
umns. In essence, each column was characterized by a specific chemical
property, such as alkali metal or chemically nonreactive gas. The atomic
134 How Math Explains the World