« ΠροηγούμενηΣυνέχεια »
And thus any proposed Number of Decimals may be turned or changed into the known Parts of what they represent, viz. Whether they be Parts of Coin, Weights, Measures, or Time, &c.
I have omitted inserting more Examples of this kind, because I take the Excellency, and indeed the chief Use, of Decimal Fractions, to consist more in Geometrical Computations, than in the common or practical Parts of Arithmetick, as will appear further on ; although even in those they are very useful upon several Accounts;, especially in the Computations of Interest and Annuities, &c. But of that more in it's proper Place. I shall therefore conclude this Chapter, with a Remark or two upon the Nature and Properties of Fractions in general.
If any given Number (whether it be whole or mixed) be multiplied with a Fraction, either Vulgar or Decimal, the Product will be less than the Multiplicand, in such a Proportion as the multiplying Fraction is less than an Unit or 1.
That is; as the Denominator of the Fraction is to it's Numerator, so will the given Number be to the Product.
Therefore, whenever any Number is to be multiplied with a Fraction, whose Numerator is an Unit: Divide that Number by the Denominator of the Fraction, and the Quotient will be the Product required. Thus 12 x 4 = 3. And 12 4= 3. Again, 12x}=6. And 12+2=6, &C.
From hence it follows, that if any Number be divided by a Fraction, the Quotient will be greater than the Dividend, by such a Proportion as Unity is greater than the dividing Fraction.
Thus 12 * = 48, viz. 5:1::12 : 48,&c. But the Truth of these will be best understood after the next Chapter.
of Continued Proportions, and how to change or vary
ibe Order of Things.
Sect. 1. Concerning Arithmetical Progression, usually called
Arithmetical Proportion Continued.
V HEN any Rank or Series of Numbers do either increase W or decrease by an equal Interval or common Difference, those Numbers are said to be in Arithmetical Progreffion,
The Number of Terms without the first is 82,
nce is : Multiply The Difference betwixt the two Extreams 16
Proposition 1. In any Series of Numbers in Arithmetical Progression, the two Extreams, and the Number of Terms being given, thence to find the Sum of all the Series.
Multiply the Sum of the two Extreams into the NumTheorem.
ber of all the Terms; and divide the Product by 2.
The Quotient will be the Sum of all that Series. Per LCorol. 1.
12 the Number of all the Terms.
EX A MPLE 2. Suppose one Hundred Eggs were placed in a Right Line a Yard distant from one another, and the first Egg were a Yard from a Basket ; whether or no may a Man gather up these 100 Eggs singly one after another, still returning with every Egg to the Basket and putting it in, before another Man can run four Miles. That is, which will run the greater Number of Yards. In this Question 200-42=202 Is the Sum of the two Extr. And
100 Is the Number of all the Terms. --
The Number of Then 2) 20200 (101003 Yards he runs that
(takes up the Eggs. Now 4 Miles=7040 Yards | The Yards he runs that takes up But 10100—7040=30602 the Eggs more than the other.
Propofition 2. In any Series of Numbers in Arithmetical Progression, the two Extreams and Number of Terms being given; thence to find the common Difference of all the Terms in that Series.
c. The Difference betwixt the two Extreams, being
J divided by the Number of Terms less than an Unit or Theorem 2.
31. The Quotient will be the common Difference of Lthe Series. Per Corol, 2.
One had Twelve Children that differed alike in all their Ages; the youngest was Nine Years old, the eldest was Thirty-fix and a half; what was the Difference of their Ages, and the Age of each? Here 36,5–9=27,5 The Difference of the two Extreams. And 12-1=li. The Number of Terms less an Unit. Then U) 27,5 (2,5 The common Difference required. Consequently 9+2,5=11,5 The Age of the youngest but one. And 11,5*2,5=1.4 The Age of the youngest but two. And fo on for the rest. Per Corol. 2.
A Debt is to be discharged at eleven several Payments to be made in Arithmetical Progression. The first Payment to be Twelve Pounds Ten Shillings, and the last to be Sixty-three Pounds. What is the whole Debt, and what must each Payment be?
Per Theorem 1. Find the whole Debt thus : 12,5+63=75,5 The Sum of the Extreams.
TI The Number of Terms, 75 5
2) 830,5 (415,25=415 l. 55. The whole Debt. Then, per Theorem 2. find the common Difference of cach Payment.
Thus 63—12,5550,5 The Difference of the Extreams.
I so . s. l. s.
1 s. l. s. l. s. And 17.11+5. 1522 . 12 The third Payment, &c.
There are eighteen Theorems more relating to Questions in Arithmetical Progression; but because they would require a great many Words to fhew the Reason of them : I therefore refer the Reader to the Second Part, viz. That of Algebra, where he may find their Analytical Investigation.
Sect. 2. Concerning Geometrical Doportion continued;
sometimes called Geometrical Progresion.
p2 . 4.8. 16. 32. &c. here 2 is the common Multiplier, 964 . 32 . 16.8.4.& c. here 2 is the common Divisor,
s 220.127.116.11. 162.& c. here 3 is the common Multiplier. "116254.18.6.2. here 3 is the common Divisor.
Note, The common Multiplier (or Divisor) is called the Ratio ; and it fhews the Habitude or Relation the Numbers have to one another, viz. whether they are Double, Triple, Quadruple, &c. which Euclid thus defines,
Ratio (or Rate) is the mutual Habitude or Respect of two Magnitudes (confequently two Numbers) of the same kind each to other, according to Quantity, Euc. 5. Def. 3:
Proportion (rather Proportionality) is a Similitude of Ratio's. Euc. 5. Def. 4.
So that there cannot be less than three Terms to form a Proportionality or Similitude of Ratio's ; and if but three Terms, the second mult supply the Place of two, As in these 2 . 4.8. That is, 2:4::4:8. (of :: see page 5.)
Here 4 the middle Term supplies the Place of two Terms, to wit, of the second and third, 8 bearing the fame Reason,