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Six Hundred Seventy Eight Thousand Millions,
Nine Hundred Eighty Seven Millions,
Six Hundred Fifty Four Thousand,

Three Hundred Twenty One Units. Of any proposed Species or Quantities whatsoever.

And here it may be observed, that every third Figure from the Place of Units, bears the Name of Hundreds; which thews that if any great Sum be parted, or rather diftinguished into Periods, of Three Figures in each Period (as in the foregoing Table) it will be of good Use to help the young Learner in the eafier valuing and expressing that Sum.

Sect. 2. Of adottion.

Poftulate or Petition. That any given Pumber may be increased or made more, by putting

another Pumber to it. addition is that Rule by which several Numbers are collected and put together, that so their Sum or Total Amount may be known.

In this Rule Two Things being carefully observed, the Work will be easily performed.

1. The first is the true placing of the Numbers, so as that cach Figure may stand directly underneath those Figures of the fame Value, viz. place Units under Units, Tens under Tens, and Hundreds under Hundreds, &c.

Then underneath the lowest Rank (always) draw a Line to separate the given Numbers from their Šum when it is found.

Example. If these Numbers 54327, and 2651, were given to be added together, they must be placed

Thus, {

S 54327

2651

2. The second thing to be observed is the due Collecting or Adding together each Row of Figures that stand over one another of the same Value: And that is thus peformed:

RULE Always begin your Addition at the Place of Units, and Add together all the Figures that stand in that Place, and if their Sum be under Ten, set it down below the Line underneath it's own Place; but if their Sum be more than Ten, you must fet dewn only the overplus, or odd Figure above the Ten (or Tens) and fa many Tens as the Sum of those Units amount to, you must carry to the place of Tens; Adding them and all the Figures that Atand in the place of Tens together, in the same manner as those of the Unics were added; then proceed in the same order to the place of Hundreds, and so on to each place until all is done. | The Sum arising from those Additions will be the Total Amount required.

EXAMPLE 1.

; Let it be required to find the Sum of the aforesaid Numbers, viz. {

54327
2651

56978 the Sum rcquired. Beginning at the place of Units, I say I and 7 is 8, which being less than 10, I set it down (according to the Rule) underneath it's own place of Units; and then proceed to the place of Tens, saying 5 and 2 is 7, which being less than 10, I set it down underneath it's own place of Tens, and proceed to do the like at the place of Hundreds, and then at Thousands, setting each of their Sums underneath their own respective places: Lastly, because there is not any Figure in the lower Rank to be added to the Figure 5, which stands in the place of Ten Thousands, in the upper Rank, I therefore bring down the said 5 to the rest, placing it underneath it's own place, and then I find that 54327+2651356978, the true Sum required.

E X A M P LE 2.

Suppose it were required to find the sum of these Numbers, 3578+496+742 +184+95. These being placed, as before directed, will ftand as in the Margin. Then beginning (as before) at the place of Units, fay 5 and 4 is 9, and 2 is it, and 6 is 17, and 8 is 25; fet down the 5 Units underneath it’s 3578 own place of Units, and carry the 20, or two Tens, to the

496 place of Tens (at which place they are only 2) saying, 2 742 : and 9 is 11, and 8 is 19, and 4 is 23, and 9 is 32, and 7 184 is 39 ; set down the 9 underneath it's own place of Tens, 2.95 and carry the 30, or three Tens (which indeed is 300) to the place of Hundreds, at which place they are buc 3, 5095 faying, 3 I carry and 1 is 4, and 7 is un, and 4 is 15, and s is 20;

here because there is no Figure overplus (as before) I set down a Cypher underneath the place of Hundreds, and carry the.. 2:Tes (or sather the 2000) to the place of Thousands, saying с

(as

(as before) 2 I carry and 3 is 5, which being the laft, I set it down underneath it's own place, and all is finished. And find the Sum or Total Amount to be 509553578+496+742+184+95.

If this Example be well confidered, it will be fufficient to fhew the usual Method of Addition in whole Numbers; but to make all plain and clear, I shall Thew the young Learner the Reason of carrying the Tens from on Degree or Row of Figures, to the next Superior Degree, which is done purely to save Trouble, and prevent the using of more Figures than are really pecelsary, as will appear by the following Method of adding together the same Numbers of the last Example.

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The Sum of the Row of Units, is
The Sum of the Row of Tens, is

Add
The Sum of the Row of Hund. is 1 olo
The three Thousand brought down 13lololo.

The Sum or Total Amount as before, is 5095

From hence I presume it will be easy to conceive the truc Reason of carrying the aforesaid Tens; and also that Cyphers do not augment or increase the Sum in Addition. (See Page 4.)

I might have here inserted a Lineal Demonstration of this Rule of Addition; but I thought it would rather puzzle than improve a young Learner, especially in this place; besides the Reason of it is fufficiently evident from that Natural Truth of the Whole being Equal to all it's Parts taken together. Euclid 1.

Axiom 19.

That is, the Numbers which are proposed to be added together, are by that Axiom understood to be the several Parts, and. their. Sum or Total Amount found by Addition is understood to be: the Whole.

And from thence is deduced the Method of proving the Truth of any Operation in Addition, viz. By parting or separating the given Numbers into Two Parcels (or more, according to the Largenefs of it) and then adding up each Parcel by it felf: For if those particular Sums so found, be added into one Sum, and that Sum prove Equal, or the same with the Total Sum first

found,

found, then all is right; if not, care must be taken to discover and correct the Error.

EXAMPLE.

5647
3289 The Sum of these Parts is, 12952

4016
Add

2900
5007 The Sum of these, is

9513 1606

The Total Sum of

these Parts.

The Sum of each
Parcel put together

Sect. 3. Of Subtraction. .

Postulate or Petition. That any number may be diminished, or made less, by taking

another Pumber from it. Subtraction is that Rule by which one Number is deducted or taken out of another, that so the Remainder, Difference, or Excess may be known.

As 6 taken out of 9, their remains 3. This 3 is also the Difference betwixt 6 and 9, or it is the Excess of 9 above 6.

Therefore the Number (or Sum) out of which Subtraction is required to be made, must be greater than (or at least equal to) the Subtrabend or Number to be subtracted.

Note, This Rule is the Converse or Direct contrary to Addition.

And here the same Caution that was given in Addition, of placing Figures directly under those of the fame Value, viz. Units under Units, Tens under Tens, and Hundreds under Hundreds, &c. must be carefully observed; also underneath the lowest Rank there must be drawn a Line (as before in Addition) to separate the given Numbers from their Difference when it is found.

Then having placed the lefler Number under the greater, the Operation may be thus performed.

RU L E. Begin at the Right Hand Figure or place of Units (as in Addition) and také or subtract the lower Figure in that place

from

C. 2

from the Figure that sands over it, setting down the Remainder or Difference underneath it's own place. If the Two Figures chance to be Equal, set down a Cypher: But if the upper Figure be less than the lower Figure, then you must add 10 to the upper Figure, or mentally call it 10 more than it is, and from that Sum fubtract the lower Figure, setting down the Remainder (as before directed). Now because the 10 thus added, was supposed to be borrowed from the next superior place (viz. of Tens) in the upper Figures, therefore you must either call the upper Figure in that place from whence the 10 was borrowed, one less than really it is, or elfe (which is all one, and most usual) you must call the lower Figure in that place one more than it really is, and then proceed to Subtraction in that place, as in the former; and so gradually an from oni Row of Figures to another until all be done.

E X A MPLE I. Let it be required to find the Difference between 6785, and 4572. That is, let 4572 be fubtracted from 6985.

These Numbers being placed down, as before directed, will stand

Thus

6785 4572

us { 4572

2213 Beginning at the place of Units, take 2 from 5 and there will remain 3 which must be set down underneath it's own place, and then proceed to the place of Tens, taking 7 from 8, and there will remain 1, to be set down underneath it's own place; again, at the place of Hundreds, take 5 from 7, and there remains 2, which let down, as before ; lastly, take 4 from 6 and there will remain 2, which being set down underneath it's own place, the Work is finished, and the Difference so found will be 2213=4785—4572, as was required.

EXAMPLE 2.
The Difference between 5849, and 7496 is required.

Having placed the Numbers as in the Margin, begin at the place of Units (as before) and say 9 from 6 cannot

7496 be, but 9 from 16 and there remains 7, to be set down 5849 under it's own place; next proceed to the place of Tens, where you must now pay the 1o that was borrowed to make the 6, 16, by accounting the upper Figure 9 in that place one less than it is, saying 4 from 8 and there remains - 4, or elle (which is the most practised) Tay 1 I borrowed and 4 is 5

from

1647

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