9.

For the readier Writing and Reading Numbers, it is now, through Cuftom, an established Rule, that, when a Number confifts of feveral Places, the Figure, which is in the first Place, on the Right-hand, fhall denote (as alfo when it ftands alone) only its fimple Value: But the Figure in the second Place, counting from the Right-hand towards the Left, fhall be so many Tens as it would be Units in the first Place; and fo onwards to the Left, always ten Times as much as the fame Figure would denote in the preceding Place. Thus, for Example Sake, in this Sum 3333, three in the first Place on the Righthand is only 3; 3 in the fecond Place towards the Left-hand is 3 Tens, or Thirty; the 3 in the third Place is ten Thirties, or three Hundred; the 3 in the fourth Place is ten three Hundreds, or 3 Thoufands: Hence that Sum may be read thus: Three Thousand, 3 Hundred, and Thirty-three.

10. But having, many Times, Occafion to write down fo many Hundreds, Thoufands, &c. without any intermediate Terms, it is neceffary, to prevent Confufion, to fill up the vacant Places with fome Character, which, in itself, fhould fignify nothing; and hence appears the Ufe of the Cipher (0) nothing; thus E. G. we may write three Hundred thus 300; for, by Virtue of the two Ciphers being placed to the Right-hand of the 3, it ftands in the third Place; and therefore, by the laft Article, denotes three Hundred. Thus alfo 30, is Thirty, and 30000, thirty Thousand; and hence it will appear that 35071 is thirty-five Thoufand and Seventy one; because the 1 ftanding in the first or Unit's Place is fimply 1; the 7 ftanding in the fecond Place is Seventy; and the 5 being in the fourth Place is five Thoufand: Laftly, the 3 being in the fifth Place is thirty Thoufand, and confequently the whole Number is thirty-five Thonfand and SeventyHence it will be no difficult Matter to underftand the following Table.

one.

11. The Learner being fuppofed to understand what has been already faid, we shall now fhew him how to read a very large Number, e. g. 614.321631.

6

4

3

543261.701810.718432. 171816. 743215. 407184.

321718.765671. The Method is thus: Over the feventh Figure, counting from the Right-hand toward the Left, put 1; from which count fix, and over it put 2, &c. as in the above Number: Then the Figure over which I ftands is Millions, that over which 2 is placed is Millions of Millions or Billions, that over which 3 ftands is Millions of Millions of Millions, or Trillions, &c. Hence the above Number may be read thus, 614 Nonillions, 321631 Otillions, 543261 Septillions, 701810 Sexillions, 718432 Quinquillions, 171816 Quadrillions, 743215 Trillions, 407184 Billions, 321718 Millions, 765671. After this Manner we may numerate any Number, though the Number of Places be very great.

12. The Writing down of Numbers being only the Reverse of Reading them, when written, we shall not here expatiate on; (especially as we fuppofe our Readers to have fome Acquaintance with more Rules than this;) but fhall only take Notice, that Youths are fometimes nonpluffed when fuch Questions as the following are propofed to them, viz. To write down in Figures 11 Thoufand, 11 Hundred, and 11; for Want

I

Want of confidering that 11 Hundred is"1 Thousand and 1 Hundred: But give them this Hint, and they will then readily know, that the above-mentioned Number is equal to 12 Thousand, 1 Hundred, and II; and then they will presently write 12111.

13. When we are to demonftrate the Truth of a Theorem that is not limited to particular Numbers, but only, that, on fuch Conditions as are therein expreffed, it will be as in the Theorem; it will not be a fufficient Proof of the Truth of the Rule, to fhew that it holds good in fome particular Numbers; and therefore we must make Use of some Contrivancé, by the Help of which we may be able to argue abstractedly from particular Numbers. In order to which, it is many Times the best Method to put fome Letters, as a, b, c, &c. to reprefent the Numbers; and then, as these Letters may ftand for any particular Numbers, whatever is deduced from Reafoning on these Letters, with the Properties common to all Numbers, must be true in all poffible Cafes; and hence the Name of Literal or Univerfal Notation This Method of putting Letters for Numbers is alfo called Algebraical Notation. We have already faid, that is the Mark for Plus; thus 2-3 is 2 Plus 3 ; and a+b is, a Plus b; alfo that is Minus, and therefore ab is a Minus b; and is equal to, as ab is a equal to b. The Signs of Comparison are and; the Mark for less than being , and for greater than; thus 23 is to be read, 2 is less than 3; and 3 2 is, 3 is greater than 2. The Sign of the Difference of two Quantities is, which is made Ufe of when we do not know which is the greatest of the Quantities; thus ab is the Difference between a and b, that is, if a be the greater Quantity, it ftands for a-b; but, if b be the greater Quantity, it is b-a. What other Characters we may occafionally make Ufe of, will be explained in their proper Places.

14. Quantities may be confidered either as affirmative, that is, greater than nothing, or negative,

B 4

viz.

viz. lefs than nothing: Thus the Cafh and Goods of a Merchant, and the Debts owing to him, may be confidered as affirmative, or greater than nothing; because they affirm he has fo much. As to negative Quantities, though to fay that any Quantity is lefs than nothing is a Contradiction, yet a Quantity may be fo circumftantiated with refpect to fome Thing, or Perfon, as to be in effect (with respect to that Thing, or Perfon) as lefs than nothing: Thus e. g. the Debts a Merchant owes, are, with refpect to him, Negative Quantities; for they are, with respect to him, really fo much less than nothing; denying him to be worth by fo much as the Debts are, what his Goods, Cash, and Debts owing to him would make him to be.

15..In Algebra is the affirmative Sign, and therefore all Quantities which have this Sign before them, are to be understood as affirmative Quantities: When there are many Quantities written, the firft, or leading Quantity, has frequently no Sign fet before it; but then it is fuppofed to have this Sign before it, or to ftand for an affirmative Quantity.

16. In the Algebraick, or Analytick Art, - before any Quantity denotes, that that Quantity is to be confidered as a negative Quantity.

17. Before we put an End to this Chapter, it will not be improper to lay down the following Corollaries.

Corollary 1. If two Numbers (written in Figures) having the fame Number of Places, that is of the greateft Value, which has the greateft Figure in the highest Place. As, for Example, 830-590; and 23602359.

18. Coroll. 2. Of two Numbers confifting of an unequal Number of Places, that is the greatest which has the greatest Number of Places, e.g. 10197.

19. Coroll. 3. If to the Right-hand of any two unequal Numbers be placed an equal Number of Figures, that will remain the greatest which was fo before.

CHAP.

20.

A

CHAP. III.

Of ADDITION.

DDITION, (from add, from the Lat. addo) teaches to find a Number equal to two or more given Numbers, taken together; that is, to find one Number called the Sum, (Summa, Lat.) or Total Sum, which fhall contain fo many Units as are contained in all the given Numbers taken together.

21. Poftulate. Grant that any Number may be increased by adding of another Number to it. See Art. 25.

22. Axiom 1. If equal Things be added to equal Things, the Sums will be equal.

23. Axiom 2. Such Quantities as are equal to one and the fame, or equal Things, are equal to each other.

24. Axiom 3. All the Parts, taken together, are equal to the Whole.

25. It is evident that the Sum of any two Num"bers may be found, by adding to one of the Numbers (the greatest is the beft) feparately, one by one, the Number of Units contained in the other Number.

Thus, for Example, the Sum of the two Digits 9 and 5 is equal to 9+1+1+1+1+1, which may be collected-together, by faying, 9+1=10, +1=11, +1=12, +1=13, +1=14. But, fince this would be a very tedious Method of adding large Numbers, we must feek out for a better; but firit it will be neceffary to make, by this Method, the following Table, expreffing the Sum of any two Digits.