In the debate of May, 1866, on Electoral Qualifications, a question arose
about arithmetical capability. Mr. Gladstone asked how many members of the
House could divide 1330l. 7s. 6d. by 2l. 13s. 8d. Six hundred and
fifty-eight, answered one member; the thing cannot be done, answered
another. There is an old paradox to which this relates: it arises out of
the ignorance of the distinction between abstract and concrete arithmetic.
_Magnitude_ may be divided by _magnitude_; and the answer is number: how
often does 12d. contain 4d.; answer three times. _Magnitude_ may be divided
by _number_, and the answer is _magnitude_: 12d. is divided in four equal
parts, what is each part? Answer three _pence_. The honorable objector,
whose name I suppress, trusting that he has mended his ways, gave the
following utterance:
"With regard to the division sum, it was quite possible to divide by a sum,
but not by money. How could any one divide money by 2l. 16s. 8d.?
(Laughter.) The question might be asked, 'How many times 2s. will go into
1l.?' but that was not dividing by money; it was simply dividing 20 by 2.
He might be asked, 'How many times will 6s. 8d. go into a pound?' but it
was only required to divide 240 by 80. If the right hon. gentleman were to
ask the hon. {249} member for Brighton (Professor Fawcett),[397] or any
other authority, he would receive the same answer--viz., that it was
possible to divide by a sum, but not by money. (Hear.)"
I shall leave all comment for the second edition, if I publish one.[398] I
shall be sure to have something to laugh at. Anything said from a
respectable quarter, or supposed to be said, is sure to find defenders. Sam
Johnson, a sound arithmetician, comparing himself, and what he alone had
done in three years, with forty French Academicians and their forty years,
said it proved that an Englishman is to a Frenchman as 40 x 40 to 3, or as
1600 to 3. Boswell, who was no great hand at arithmetic, made him say that
an Englishman is to a Frenchman as 3 to 1600. When I pointed this out, the
supposed Johnson was defended through thick and thin in _Notes and
Queries_.
I am now curious to see whether the following will find a palliator. It is
from "Tristram Shandy," book V. chapter 3. There are two curious idioms,
"for for" and "half in half"; but these have nothing to do with my point:
"A blessing which tied up my father's tongue, and a misfortune which set it
loose with a good grace, were pretty equal: sometimes, indeed, the
misfortune was the better of the two; for, for instance, where the pleasure
of harangue was as _ten_, and the pain of the misfortune but as _five_, my
father gained half in half; and consequently was as well again off as if it
had never befallen him."
This is a jolly confusion of ideas; and wants nothing but a defender to
make it perfect. A person who invests five {250} with a return of ten, and
one who loses five with one hand and gains ten with the other, both leave
off five richer than they began, no doubt. The first gains "half in half,"
more properly "half _on_ half," that is, of the return, 10, the second 5 is
gain upon the first 5 invested. "Half _in_ half" is a queer way of saying
cent. per cent. If the 5l. invested be all the man had in the world, he
comes out, after the gain, twice as well off as he began, with reference to
his whole fortune. But it is very odd to say that balance of 5l. gain is
_twice_ as good as if nothing had befallen, either loss or gain. A
mathematician thinks 5 an infinite number of times as great as 0. The whole
confusion is not so apparent when money is in question: for money is money
whether gained or lost. But though pleasure and pain stand to one another
in the same algebraical relation as money gained and lost, yet there is
more than algebra can take account of in the difference.
Next, Ri. Milward[399] (Richard, no doubt, but it cannot be proved) who
published Selden's[400] Table Talk, which he had collected while serving as
amanuensis, makes Selden say, "A subsidy was counted the fifth part of a
man's estate; and so fifty subsidies is five and forty times more than a
man is worth." For _times_ read _subsidies_, which seems part of the
confusion, and there remains the making all the subsidies equal to the
first, though the whole of which they are to be the fifths is perpetually
diminished.
Thirdly, there is the confusion of the great misomath {251} of our own day,
who discovered two quantities which he avers to be identically the same,
but the greater the one the less the other. He had a truth in his mind,
which his notions of quantity were inadequate to clothe in language. This
erroneous phraseology has not found a defender; and I am almost inclined to
say, with Falstaff, The poor abuses of the time want countenance.