Saint Mary's beacon. [volume] (Leonard Town, Md.) 1867-1983, 1875-09-16

be had become able % ~ -
others the methods ~ w |D
were obtained, . real a6vl -
edge might have been lse1 -
il seems TO have been ~
him, = with George Bi11 ~
scaIculating boys that VI -
culture Of the mind til g -
faded away"
With all respect for = - -
% competent % Judge q -
. Fnncis Baily, ~ - -
PLANTATION seems a1toGoehee TT
So far from the properties = -
illustrating The fsa1D = =
they illnstzA&e The Tq s
isa1 1ovooepunen6el 0 L.
posits direction. The sys1G9 q loga-
rithms enables the calculates 10 obtain
results more quickly than Of old, mot BY
the more active exercise Of hi own pow
ers Of calculation. but BY employing re-
suits accumulated by the labors Of 9th
ers. Its great advantage, aud the qual-
ity which causes every mathematician
to be grateful to the memory of Napier
of Merchistoun, resides in the fact that,
by taking advantage Of . well-known
property Of numbers, tables of moder-
ate dimensions serve Q great number Of
purposes which by any ordinary plan of
tabulation, would require several vol
umes of great size. If it were possible
for calculator to use as readily q set
Of tables equal in bulk to five volumes
of the ''London Directory" as he now
uses q book of logarithms, and if such
volumes could be as easily and as cheap
ly produced, tables much more labor.
saving than the books Of logarithms
could be constructed. But Of course
such sets of volumes would be practi-
cally useless if they could be produced,
and it would be impossible either to find
calculators to form the tables or printers
and publishers to bring them out. Now,
of all processes BY which mathematical
calculation can be carried out, no two
can be more unlike than mental arith-
metic on the one hand, and the use of
tables, Of whatever kind, on the other.
Napier invented his system to reduce as
far as possible the mental effort in cal
culation, making the calculator employ
results collected by others; young Col
burns success depended on mental
readiness; and he was 30 far from us
ing the results obtained by others, that
he did not even know the ordinary
methods of arithmetic. A man of Ma
piers way of thinking would be the last
to trust to mental calculation; whereas,
If Colburn had retained his skill until
he had acquired power to explain his
methcas hs would have beep the last to
think of such a help to SssJ6iation as q
table of logarithms. Napier strongly
urged the advantage Of aids to calcula-
tion; Colburn would scarcely have been
able to imagine their necessity.
Nor is it at all ikely-we could even
say it is not possible-that properties
of numbers exist through the knowl-
edge of which what Colburn did could
be commonly done. The mathemati-
clans who have dealt with theory of
numbers have been too numerous and
too skillful, and worked too diligently
in their field of research, to overlook
such properties, if they existed. Be-
sides, it is scarcely reasonable to sup
pose that q child who had but lately
learned the nature of numbers, and was
altogether unacquainted with the ordi-
nary properties, should have intuitive-
ly recognized abstruser properties. A
more natural explanation must surely
aviSf ~ oe1ar <hn k%n, okkc,
lively.
1t happens that the writer is able,
from his own experience, to advance an
explanation which accords well with the
facts, and especially with the circum-
stance that calculating boys usually lose
their exceptional power of rapid reck-
oning when they are instructed in and
taught to practice the ordinary meth
DDS; for the writer used formerly to
possess, though in q slight degree only,
a power of finding divisors, products,
and so on, which-uplike ordinary skill
in calculation-required only to be ex-
panded to effect what Colburn effected.
1t was, in point of fact, simply the pow
er of picturing a number (not the writ
ten number, but SO many ''thiugs"), },
and changes in the number, correspond
ing to division or multiplication as the
case might be. Thus the number 24
would be presented as two columns of
dots each containing ten, and one COL
umn containing four on the right
of the columns of ten. If this num-
Der were to be multiplied by three,
all that was necessary was to picture
three sets of dots like that Just de-
scribed; then to conceive the imperfect
columns brought together on the right,
giving six columns of ten and three col
umns each of four dots; and these
three gave at once (by heaping them
up properly another column of ten
with two over; in all seven columns of
ten and one column of two-that is 72.
This takes long i'm writing, but, as
pictured in the minds eve, the three
sets representing "a formed themselves
into a single set representing 72 in the
twinkling of an eve (f the minds eve
can be imagined twinkling). The pro-
cess for division was not exactly the re.
verse Of that for multiplication. Thus,
~ being pictured rs seven columns Of
ten and one Of two, to divide it By 8,
the first six columns Of ten were pic
tured rs giving twenty sets of three
horizontal dots; the next column of ten
gave three vertical triplets, counted
from the top; and then the remaining
dot at bottom, with the other two in
the imperfect column, gave another
triplet, or twenty-four triplets in ALL
These tripleis could all be seen as it
were; and the only mental calculation