As to the Reafon (or Proof) of this Rule for dividing Fractions: It is only the Converse to that of Multiplication, and will be very evident from this following.

Let be divided by . Which according to the Rule is thus, ) (24. The true Quotient. Now 8. And 2. per Sect. 3. Confequently divided by is but the fame with 8 divided by 2. viz. 2.) 8 (4. The Quotient as

before.

I could have inferted Geometrical Demonftrations, for the Rules of Multiplication and Divifion of Fractions; but fuppofing the Learner purely unacquainted with thofe kind of Demonftrations, I thought these might be more intelligible to him, efspecially in this place.

CHA P. V.

Of Decimal Fractions.

WHEN,

or by whom, this excellent Invention of Decimal Arithmetick, was firft introduced is uncertain; but doubtless it's Improvements, and the Perfections it is now in, is owing to later

Years.

Sect. 1. Of Motation.

I N Decimal Fractions, the Integer or whole Thing (whether it be Coin, Weight, Meafure, or Time, &c.) is fupposed to be divided into Ten equal Parts; and every one of those Ten Parts are fuppofed to be fubdivided into other Ten equal Parts, &c. ad infinitum.

The Integer being thus divided (by Imagination) into 10, 100, 1000, 10000, &c. equal Parts, becomes the Denominator to the Decimal Fractions.

2

33

Thus, 1703. 18300. 12546, &c.
730 100000,

Now thefe Denominators are feldom or never fet down, but only the Numerators; and thofe are either diftinguished, or fepa rated from whole Numbers by a Point, or a Comma.

Thus, 5,4 is 5 . and 0,7 is . 35,05 is 35 %, &c.

But before we proceed further in Notation, it will be conveni ent for the Learner to confider the following Table, (taken out of the learned Mr Oughtred's Clavis Mathematica) which fhews the very Foundation of Decimal Fractions.

I

Whole

By this Table it is evident, that as in the whole Numbers or Inte gers, every Degree from the Units Place increases towards the left-hand by a Ten-fold Proportion: So in Decimal Parts every Degree is decreased towards the right-hand by the fame Proportion, viz. by Tens.

Therefore thefe Decimal Parts or Fractions, are really more Homogeneal, or agreeing with whole Numbers, than Vulgar Frac tions; for indeed all plain Numbers are in effect but Decimal Parts one to another.

That is, fuppofe any Series of equal Numbers, as 444, &c. 'The first 4 towards the Left is Ten times the Value of the 4 in the middle, and that 4 in the middle is Ten times the Value of the last 4 to the Right of it, and but the Tenth Part of that 4 on the Left, &c.

Therefore all or any of them may be taken either as Integers, or Parts of an Integer: If Integers, then they must be fet down without any Comma or feparating Point betwixt them thus, 444. But if Integers, and one Part or Fraction, put a Comma betwixt them thus, 44,4 which fignifies 44 whole Numbers, and 4 Tenths of an Unit: Again, if two Places of Parts be required, feparate them with a Comma thus, 4,44 viz. 4 Units, and 44 hundred Parts of an Unit, &c.

From hence (duly compared with the Table) it will be eafy to conceive that Decimal Parts take their Denomination from the Place of their laft Figure.

Cyphers annexed to Decimal Parts, alter not their Value. As ,50,,500, or ,5000, &c. are each but 5 Tenths of an Unit, For. And

of the laft Chapter.

500

1800

ic. Or 5000

10000

1

Per Sect. 4.

But Cyphers prefixed to Decimal Parts decrease their Value, by removing them further from the Comma.

Thus,

,55 Tenth Parts.

,055 Parts of a Hundred.

,0955 Parts of a Thousand.

,00055 Parts of Ten Thousand, &c.

Confequently the true Value of all Decimal Parts are known by their Distance from the Units Place; the which being once rightly. understood, the reft will be eafy.

Sect. 2. Addition and Subtraction of Decimals.

IN

N fetting down the propofed Numbers to be added, or subtracted, great care muft be taken in placing every Figure directly underneath those of the fame Value, whether they be mixed Numbers, or pure Decimal Parts, and to perform that you must have a due regard to the Comma's, or feparating Points, which ought always to ftand in a direct Line one under another; and to the Right-hand of them carefully place the Decimal Parts, according to their refpective Values, or Diftances from Unity. Then

Add or fubtract them, as if they were all whole Numbers ; Rule and from their Sum, or Difference, cut off so many Decimal Parts as are the most in any of the given Numbers.

EXAMPLES in Addition.

Let it be required to find the Sum of these following Numbers, viz. 34,5 + 65,3 + 128,7 +95 + 87,8 +7,9, which being truly placed, will stand

EXAMPLE 2.

Let it be required to find the Sum of 25,854+34,578+9,076 +13,907.

25,854
34,578

9,076

13,907

83,415 The Sum required.

When the Decimal Parts propofed to be added (or fubtracted) have not the fame Number of Places, you may for convenience of Operation supply or fill up the void Places, by annexing Cyphers. As in these Examples.

EXAMPLE 3. EXAMPLE 4.

45,0700

EXAMPLE 5.

50,7580

74,284.

EXAMPLE 1.

Let it be required to find the Difference between 45,375 and

EXAMPLE 2. EXAMPLE 3.

That is, From 74,284

From 437,5

From 75,0034

Take 45,375

Take 89,657

Take 57,875

Remains 28,909

347,843

17,1284

EXAMPLE 4.

Let it be required to find the Excefs between 562 and 93,5784.

Note, The two laft Examples are fuppofed to be fupplied with

Cyphers, which if actually done would stand thus,

0,108135

The Proof of Addition and Subtraction in Decimals, is the fame with that of whole Numbers, page 13, &c.

Sect. 3.

Multiplication of Decimals.

WHETHER the Factors or Numbers to be multiplied are

pure Decimals, or mixed. Multiply them as if they were all whole Numbers, and for the true Value of their Product obferve this Cut off (viz. feparate with a Comma) fo many Places of Rule. Decimal Parts in the Product, as there are in both the Factors accounted together. As in thefe.

EXAMPLE

3,024

I.

EXAMPLE

32,12

2.

6,743 52

780,516

The Reason why fuch a Number of Decimal Parts must be cut off in the Product, may be easily deduced from these Examples. Thus,

In Example 1. It is evident, that 3, the whole Number in the Multiplicand, being multiplied with 2, the whole Number in the Multiplier; can produce but 6 (viz. 3 x 26). So that of neceffity all the other Figures in the Product must be Decimal Parts; according as the Rule directs.

Or, the Rule is evident from the Multiplication of whole Numbers only: Thus, fuppofe 3000 were to be multiplied with 200, their Product will be 600000; That is, there will be so many Cyphers in the Product, as are in both the Factors, (Vide page 18.) Now if, inftead of thofe Cyphers in the Factors, we fuppofe the like Number of Decimal Parts; then it follows, that there ought to be the fame Number of Decimal Parts in the Product, as there were Cyphers in the Factors.

Again, the Rule may be otherwise made evident from Vulgar Fractions, thus: Let 32,12 be multiplied with 24,3,

and