Calculating tables giving the products of every two numbers from one to one thousand and their application to the multiplication and division of all numbers above one thousand

1897·24.1 MB·English

by  Crelle, A. L. (August Leopold), 1780-1855

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DR. CRELLE'S
A. L.
CALCULATING TABLES
GIVING
THE PRODUCTS OF EVERY TWO NUMBERS FROM ONE
TO ONE THOUSAND
AND
THEIR APPLICATION TO THE MULTIPLICATION AND DIVISION
OF ALL NUMBERS ABOVE ONE THOUSAND
REVISED BY DR. C. BREMIKER
FIRST ENGLISH EDITION
LONDON: DAVID NUTT, 270-271, STKAND
BEELIN: EEIMEE
G.
1897
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DIRECTIONS FOR USIN^G THE TABLES
These tables contain the product of every two numbers from I to 1000. They thus form
an extended multiplication table of all numbers up to 1000, just as the ordinary table gives
those of the simple numbers.
The use of the tables is as follows
:
When
in multiplication the factors, and in division the divisor and the quotient, are
each below 1000, these tables save all calculation* for they give at sight the products of
these numbers. They also simplify calculation with numbers above 1000, in proportion to
the number of figures that can be taken together.
For example, to find the product of two numbers of six figures each with the ordinary
multiplication table, it is necessary to add together the products of each figure of the
multiplicand with each figure of the multiplier, in all 36 products, whereas with these
tables, by taking each number of 6 figures in two sets of 3, and treating each set as a
simple number, only 4 products are needed, thus reducing the number of products to be
added together to one-ninth. In division, also, there is the same saving of
labour.
It is of course unnecessary to use these tables for the products of simple numbers
which are learnt by heart in the ordinary multiplication table, but in all higher calcu-
lations, the looking-up of products must be regarded as a very trifling matter compared
with the labour of ordinary processes and then these tables save labour. Logarithms,
;
so much used in the multiplication and division of high numbers, involve the same
looking-up in tables, but they cannot be used for all numbers whatever, but only for
those of 6, or at most of 8 figures, and even then the results are only approximate, and
not exact as with our tables.
In cases where there is need of various products of two factors, as for instance in
the calculation of areas of plane geometrical figures by means of triangles, which often
DIEEOTIONS FOR USING THE TABLES
iv
occurs in land-measuring and in the Ordnance Survey and Inland Revenue Offices, these
extended tables save a great amount oflabour.
They can also be used to advantage instead of logarithms, in many other mathematical
calculations. It is for instance often necessary to transform formulae in order to adapt
them to logarithmic computation. Now sometimes these transformations are not practicable
or at least not easy, as in polygonometric formulae, and then, as a consequence,
logarithms cannot be used. Now by making multiplication and division easier these
transformations become unnecessary, the formulae can be used just as they are, and
calculation is simplified. In ordinary life in every case of high numbers our tables diminish
labour.
A second advantage, as great as, ifnot greater than, that of economy of labour, is the
assured accuracy afforded by these tables. The chances of error in a computation certainly
diminish in proportion to the number of operations involved in the result. For example,
if in the aforesaid multiplication of 2 numbers of 6 figures each, the addition is required
of only 4 products found in the tables, the possibility of making a mistake is obviously 9
times less than when for the same calculation 36 products are necessary, worked mentally
—
before putting on paper. The conviction that one is calculating more surely a conviction
—
that logarithmic computation does not give is of undoubted importance. Any one who
has had to calculate with high numbers knoAvs how unsatisfactory it is, not to be certain
at every step that the work is right. These tables increase this certainty, and in this
respect possess an advantage over methods which in certain cases may be easier to work,
though in their nature they are more complicated.
The arrangement of the tables is as follows Each page contains 2000 products, namely
:
of the 2 numbers in large type at the top with all numbers from 1 to 1000.
Of the 2 factors, one is the number in large figures at the top of the page ; of the other
factor, the hundreds are on the same horizontal line as the first factor, and the tens and
the units in the same vertical column. By the side of this column, which contains all
numbers from 1 to 100, are the products of these numbers and the number at the top of
the page in the next column are the products of these same numbers increased by 100.
;
In the same manner each succeeding column represents an increase of 100, so that the
last column but one contains the products of the number at the top of the page and the
numbers of the first column increased by 900. It is necessary to remark that the last two
figures of the columns above which are 100, 200, &c., to 900, have been omitted a.nd
placed apart in the last column, since they are the same for all products given on the
same line.
Thus, to obtain the product of 439 by 382, find the page at the top of which is the
number 439, or that of the number 382. In the former case look in the top line for 300,