MORE EASY INTRODUCTORY REMARKS TENDING TO THE UNDERSTANDING OF THE COPIOUS AND IMPORTANT RULE OF

REDUCTION.

Before proceeding with reduction, a work which has cost the author much labour, having never lost sight of the great principle he has discovered from the time he first conceived the project of bringing it to perfection, the reader will naturally expect to be in some degree prepared for the task he is about to enter upon.

The author, therefore, begins with the numeration table; but, as there are various tables of the kind, which answer no useful purpose, he here inserts the one he most approves of.

In the above table the figures shew how many digits must be taken to express thousands, tens of thousands, &c.

Thus, if asked to write a million, first write down an unit and six ciphers, because, 1 added to 6 is 7; which answer to millions by the table.

The reader should always make it a rule, whenever he hears a person speak of numbers, to conceive in his mind a comma, to mark the distinction in the value of the figures;

thus, for example: Suppose a person say 87 millions, &c. Here he first expresses tens of millions; and, immediately it is easily imagined, as if in plain legible figures, with a comma on the right of the 87, and it may proceed till he has concluded with a comma on the right of every third figure from left to right; something in manner following, thus, 87,684,347.

But it requires 12 figures to express the highest number of millions, namely, hundreds of thousands of millions; add six figures more, and we have the highest number of billions; add six more again, will make 24 figures; this will express the highest number of trillions; and so on by adding six every time for the highest number of the next superior.

If 7 figures are added to 6, they must be expressed with the units' place of the billions.

If, from 24 figures, 7 of them are taken away, there will then be left 17 figures; and as 12 is required to express the highest number of millions, there will then be left 5 figures for billions; which is tens of thousands of billions, as 17,514, &c.

The reader will do well, occasionally, to practise such exercises mentally, as the above.

6

SUBTRACTION.

This rule being of great importance in reduction, the reader will do well to make himself master of the methods laid down; and, in order to render a knowledge of this as clear as possible, it will be necessary that attention be paid to the following methods.

From 768 take 345.

Here there can be no difficulty in giving the answer, because the figures to be deducted are less than the greater sum.

As it often happens that some of the figures in the subtrahend are greater than those the reader has to subtract from, it may here be as well to give him an example.

From 1719 take 984.

Here it will be seen by subtracting, that 7 for hundreds will be the left hand figure; and the other two are thus obtained; take 84 (the two right hand figures of the last

number) from 100, and there remains 16, which immediately add to 19, (the two right hand figures of the greatest number,) and the sum is 35; so that 735 is the difference required. This, by frequent practice, will become quite easy.

EXAMPLES FOR PRACTICE.

From 876 take 798.
Answer, 78.
What is the difference between 904 and 737 ?

Answer, 167.

;

The next thing to be pointed out is the method of subtracting when there are any number of figures, however large, which is done by pointing off the figures by threes and so subtracting one period from another, we will here number every period, so that the numbers of one shall answer to the numbers of the other; which are to be deducted from the figures of the sum whose value is greatest.

EXAMPLE.

From 76,843,870 take 67,916, 730

First glance your eye to the left hand figure of number 2, on the right, which is 9; this being more than the left hand figure of number 2 on the left, namely, the 8; and as an unit would be required to be carried from thence, say 67+1=68, which being taken from 76, the remainder is 8 for millions, thus, number 1 period of both sides is finished

Now for number 2 points; look to the right hand period of number 3 of the sum to be deducted, which is found to be less than the left hand figure of number 3, on the opposite side; therefore, nothing is to be added; take then 916 from 843, or rather from 1843, and there remains 927 for thousands.

Lastly, take 730 from 870 and there remains 140.

So that the difference is 8,927,140.

The above method, although it may appear tedious at first, will, nevertheless, by practice, become exceedingly easy.

THE APPLICATION OF NUMERATION IN THE

REDUCING OF MILES TO BARLEY CORNS.

When a person can, in a moment, calculate the number of barley corns in several miles, he is esteemed a clever ingenious person.

The following having come under the author's notice, he believes it deserving a place here; notwithstanding he has treated on the method of reducing miles to barley corns by the means of money; but the following observations, nevertheless, may not be amiss.

There are 190,080 barley corns in a mile.

Here are hundreds of thousands, which, when multiplied by any figure less than six, the number of figures in the product will not be altered; they will still be hundreds of thousands. It will, therefore, only be necessary to remember the two left hand figures, namely, 19; and the two right hand figures, namely, 80, to give an answer in a moment; thus, for example:

How many barley corns in 5 miles ?

Here 19X5=95

Now, suppose à cipher on the right of these, and we have 950 for thousands.

And 80×5400.

The answer, therefore, is 950,400..

But if we multiply by 6, we shall have a figure more in the product; consequently, we shall have millions; and the operation will be performed thus, 6 times 19 is 114; the left hand figure 1 is millions; and, consequently, the answer is 1,140,480.

And in this manner you may proceed to find, in a moment, the barley corns to the extent of 12 miles; because the two right hand figures 80, multiplied by 12, do not affect the units' place of the thousands, so that it will always be a cipher in the thousandth's place; but, if multiplied by 121, it will alter the product, because 123 times 80 is 1000; so that for every 12 miles 1 must be added to the unit's place of the thousands; for 25, two; from 25 to 37, three; from 38 to 49, four; and so on in like manner, must be added; and if the multiples of 19 be learnt by heart, the reader may, with little practice, find the barley corns that will reach a great number of miles.

Note. By first bearing in mind what the figure in the unit's place of the thousands ought to be, will enable the reader to finish his periods. When verbally giving his answer, he has then only to remember to bring in the three last figures to finish.

EXAMPLES.

How many barley corns are there in 14 miles?

Here 19×14=2,66.

Put an unit to the right of these and we have 2,661.
And 80×14=1,120

Omit the unit in the thousandth's place (because it has
been annexed to the former figures) and we have
2,661,120 for the answer.

How many barley corns in 54 miles?

Here 19 × 54 (or 9 times 6)=10,26.

Add 4 to the right of these we have 10,264.

And 80 X 54=4,320

The 4 being omitted (because it has been annexed to the 10,26) the answer is 10,264,320.

METHOD OF REDUCING ANY NUMBER OF

POUNDS TO FARTHINGS VERY EXPE-
DITIOUSLY BY THE HEAD, OR
MENTALLY.

Before proceeding to give instructions for reducing pounds sterling to farthings, it may be necessary to say something of its merits.

1st. It enables the possessors of this branch of knowledge to call out, at once, the highest value of his product; and so by continuing on with three figures from left to right, as in the numeration table, end his answer. Thus :

If asked how many farthings are in £76,845?

Here are tens of thousands, which, by the author's method, are known to be tens of millions of farthings

AGAIN.

How many farthings in £876,845 ?

Here there are hundreds of thousands of pounds, which answers to hundreds of millions of farthings; but that the reader may have a better knowledge of the