in the plot of a problem are three people
called A, B, and C. The form of the question is generally
of this sort:

"A, B, and C do a certain piece of work. A can do as much
work in one hour as B in two, or C in four. Find how long
they work at it."

Or thus:

"A, B, and C are employed to dig a ditch. A can dig as
much in one hour as B can dig in two, and B can dig twice
as fast as C. Find how long, etc. etc."

Or after this wise:

"A lays a wager that he can walk faster than B or C. A
can walk half as fast again as B, and C is only an
indifferent walker. Find how far, and so forth."

The occupations of A, B, and C are many and varied. In
the older arithmetics they contented themselves with
doing "a certain piece of work." This statement of the
case however, was found too sly and mysterious, or possibly
lacking in romantic charm. It became the fashion to define
the job more clearly and to set them at walking matches,
ditch-digging, regattas, and piling cord wood. At times,
they became commercial and entered into partnership,
having with their old mystery a "certain" capital. Above
all they revel in motion. When they tire of
walking-matches--A rides on horseback, or borrows a
bicycle and competes with his weaker-minded associates
on foot. Now they race on locomotives; now they row; or
again they become historical and engage stage-coaches;
or at times they are aquatic and swim. If their occupation
is actual work they prefer to pump water into cisterns,
two of which leak through holes in the bottom and one of
which is water-tight. A, of course, has the good one; he
also takes the bicycle, and the best locomotive, and the
right of swimming with the current. Whatever they do they
put money on it, being all three sports. A always wins.

In the early chapters of the arithmetic, their identity
is concealed under the names John, William, and Henry,
and they wrangle over the division of marbles. In algebra
they are often called X, Y, Z. But these are only their
Christian names, and they are really the same people.

Now to one who has followed the history of these men
through countless pages of problems, watched them in
their leisure hours dallying with cord wood, and seen
their panting sides heave in the full frenzy of filling
a cistern with a leak in it, they become something more
than mere symbols. They appear as creatures of flesh and
blood, living men with their own passions, ambitions,
and aspirations like