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Collectanea
Reductions of Decimals
Case 1
To reduce a vulgar
fraction into a decimal
Rule
Annex cyphers to
the numerator, till it
be equal to, or greater
than the denominator;
then divide by
the denominator,
and the quotient will
be the decimal sought.
Case II
To reduce coins,
weights, measures &c
into decimals
Rule
Reduce the different
species into one, viz:
the lowest denomination
they consist of
for a dividend; then
reduce the integer into
the same denomination
for a divisor;
the result will be the
decimal required.
Case III
To reduce any decimal
into the equivalent
known parts of coin,
weight, or measure
Rule
Multiply the given
number by the number
of units contained in
the next inferior
denomination, cutting
off as many figures
from the product as
the given decimal consists
of; then multiply
the remaining
parts (if any) by the
next lower denomination,
cutting off as
before; and thus
proceed till you have
converted your decimals,
or come to the
lowest part; and
the several figures
to the left hand of
the separating points
will be the several
parts of the quantity
required.
Case IV.
To reduce a decimal
into its least equivalent
vulgar fraction.
1st. If the decimal be
finite.
Rule.
Under the given decimal
write an unit, with
as many cyphers as
the decimal consists of
places; then divide both
the numerator and denominator
by the
greatest common measure,
which gives the
least equivalent vulgar
fraction required.
2nd. If the given decimal
be a repetend,
Rule.
The decimal is the numerator
of a vulgar
fraction, whose denominator
consists of as
many nines as there
are recurring places
in the given decimal;
both which divide by
their greatest common
measure, and their
quotient will be the least
equivalent vulgar
fraction.
3rd When the given decimal
is part final,
and part a circulate,
Rule.
To as many nines
as there are figures
in the repetend, annex
as many cyphers
as there are
finite places for a
denominator; then
multiply the nines
in the said denominator
by the finite
parts, and to the product
add the repeating
decimal for a numerator;
these divided by
their greatest common
measure, will give the
least equivalent fraction.
Addition of Decimals.
Case. I.
To add finite decimals.
Rule.
Add as in whole
numbers, and from
the sum or difference,
cut off so many places
for decimals, as
are equal to the greatest
number of decimal
places in any
of the given numbers.
Case. II.
To add decimals
wherein are single
repetends.
Rule.
Make every line end
at the same place,
filling, up the vacancies
by the repeating
digits, and annexing
a cypher or cyphers
to the finite
terms; then add as
before, only increase
the sum of the right
hand row with as
many units as it
contains nines; and
the figure in the sum,
under that place,
will be a repetend.
Subtraction of
Decimals.
Case I.
To subtract finite
decimals.
Rule.
Having first set
down the greater of
the two numbers given,
set down the less
under it, then subtract
as in whole numbers.
Case. II.
To subtract decimals
that have repetends.
Rule.
Make the repetends
similar & conterminous,
and subtract
as in the last case:
observing only, if the
repetend of the number
to be subtracted,
be greater than the
repetend of the number
it is to be taken
from, then the right
hand figure of the
remainder must be
less by unity, than it
would be, if the expressions
were finite.
Multiplication
of Decimals.
Case. I.
When both factors are
finite decimals, whether
they are single,
or joined with integers,
Rule.
Multiply them as
if they were all whole
numbers, and from
the product (towards
the right hand) cut
off so many places
for decimal parts in
the product, as there
were in both the multiplier
& multiplicand
counted together.
But if it so
happen that there
are not so many
places in the product,
supply the
defect by prefixing
cyphers.
Case II.
Two decimal fractions
being given, to reserve
in their product
any assigned number
of places.
Rule.
Set the unit's place
of the multiplier directly
under that
figure of the decimal
part of the multiplicand,
whose place
you would reserve
in the product, and
invert the order of
all its other places;
that is, write the decimals
on the left
hand, and the integers,
if any, on the
right.
Then in multiplying,
always begin at that
figure of the multiplicand
which stands
over the figure wherewith
you are then
multiplying, setting
down the first figure
of each particular
product underneath
one another, due
regard being had
to the increase which
would arise out of
the two next figures
to the right hand of that
figure in the multiplicand,
which you then begin with,
carrying one from 5 to 15;
two from 15 to 25; three
from 25 to 35, &c. and the
sum of these lines will give
the product.
Case III.
If the right hand figure
of the multiplicand be a
circulate,
Rule.
In multiplying increase
the right hand figure of
each resulting line by as
many units as there are
nines in the product of
the first figure in that
line, and the right hand
figure of each line will be
a circulate; and before
you add them together,
make them all end at
the same place
Case IV.
If the right hand figure
of the multiplier be a
circulate,
Rule.
Multiply by it as by a
finite digit, setting the
product one place extraordinary
towards the
left hand; then divide
that product by 9, continuing
the quotient (if
needful) till it arrives at
a circulate; then beginning
at the place under
the right hand figure of
the multiplicand, cut off
for decimal parts.
Case. V.
When the multiplicand
and multiplier are each
a single circulate,
Rule.
The first line (or that
produced by multiplying
by the circulate in
the multiplier) must be
managed as in case III,
only the right hand figure
must be encreased
by as many units as
there are nines in the
product of the first figure
of that line, the
products of the rest
must be managed as
directed in Case II.
Case VI.
If the multiplicand be
a compound repetend;
and the multiplier a
finite number,
Rule.
Identifier:  JB/135/088/002 "JB/" can not be assigned to a declared number type with value 135.



135 
posology 

088 
collectanea 

002 
reduction of decimals / addition of decimals / subtraction of decimals / division of decimals 

collectanea 
4 

recto 
[[page_numbering::f153 / / f152 [sic] /]] 

46206 
