And thus any propofed Number of Decimals may be turned or changed into the known Parts of what they reprefent, viz. Whether they be Parts of Coin, Weights, Measures, or Time, &c. I have omitted inferting more Examples of this kind, because I take the Excellency, and indeed the chief Ufe, 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 feveral Accounts; especially in the Computations of Intereft and Annuities, &c. But of that more in it's proper Place. I fhall 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 lefs than the Multiplicand, in fuch a Proportion as the multiplying Fraction is lefs than an Unit or 1. That is; as the Denominator of the Fraction is to it's Numerator, fo will the given Number be to the Product. Therefore, whenever any Number is to be multiplied with a Fraction, whofe Numerator is an Unit: Divide that Number by the Denominator of the Fraction, and the Quotient will be the Product required. Thus 12 x 43. And 1243. Again, 12x=6. And 1226, &c. From hence it follows, that if any Number be divided by a Fraction, the Quotient will be greater than the Dividend, by fuch a Proportion as Unity is greater than the dividing Fraction. Thus 12÷ 48, viz. ‡: 1 :: 12: 48, &c. But the Truth of these will be beft understood after the next Chapter. = CHAP VI. Of Continued Proportions, and how to change or vary the Order of Things. Sect. 1. Concerning Arithmetical Progreffion, ufually called Arithmetical Proportion Continued. WHEN any Rank or Series of Numbers do either increase or decrease by an equal Interval or common Difference, thofe Numbers are faid to be in Arithmetical Progreffion. 13. &c. } } Difference is 2. And fo of any other Series, whofe common Difference is 5. &c. Lemma 1. If any three Numbers be in Arithmetical Progreffion, the Sum of the two Extreams (viz. the first and last) will be equal to the Double of the Mean or middle Number. As in thefe, 2.4.6. Or 3.6.9. Or 3. 7. 11. Viz. 2+6=4+4. Or 3+9=6+6. And 3+11=7+7. &c. Lemma 2. If any four Numbers are in Arithmetical Progreffion, the Sum of the two Extreams will be equal to the Sum of the two Means. As in these, 2 4. 6.8. Or 3.6 9 12. Viz. 2+8=4+6. And 3+12=6+9. &c. Corollary. 1. From these two Lemma's it is eafy to conceive, that if never fo many Numbers be in Arithmetical Progreffion, the Sum of the two Extreams will be equal to the Sum of any two Means, that are equally diftant from thofe Extreams. As in these, 2 4 6 8 ΤΟ 12 Or if the Number of Terms be odd, as these, 2.4 6. 8. 10. 12 14. 16. 18. &c. Then 2+18=4+16=6+14=8+12=10+10. Lemma 3. Every Series of Numbers in Arithmetical Progreffion is compo fed of the Interval or common Difference, fo often repeated as there are Terms in the Progreffion, except the firft. As in thefe, 1. 3. 5. 7. 9. 11. 13. 15. 17. &c. Here the Interval or common Difference being two, it will be 1+2=3. 3+2=5.5+2=7. 7+2=9. 9+2=11. 11+2=13. 13+2=15. 15+2=17. &c. Corollary 2. Hence it is evident, that the Difference betwixt the two Extreams (viz. 1 and 17) is compofed of the common Difference, multiplied into the Number of all the Terms, excepting the first. As in the aforefaid Progreffion, 1. 3. 5. 7. 9. 11. 15. 176 The Number of Terms without the first is 8 The common Difference is Multiply The Difference betwixt the two Extreams Propofition 1. 16 In any Series of Numbers in Arithmetical Progreffion, the two Extreams, and the Number of Terms being given, thence to find the Sum of all the Series. Theorem. Multiply the Sum of the two Extreams into the Num Sber of all the Terms; and divide the Product by z The Quotient will be the Sum of all that Series. Per It is required to find the Number of all the Strokes a Clock ftrikes in one whole Revolution of the Index, viz. twelve Hours. Here 1+12=13 the Sum of the two Extreams. the Number of all the Terms. Then 2) 156 (78. The Number of Strokes required. EXAMPLE 2. Suppofe one Hundred Eggs were placed in a Right Line a Yard diftant from one another, and the firft Egg were a Yard from a Basket; whether or no may a Man gather up thefe 100 Eggs fingly one after another, ftill 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+2=202 Is the Sum of the two Extr. 100 Is the Number of all the Terms. And Then 2) 20200 (10100 Now 4 Miles 7040 Yards S The Yards he The Number of Yards he runs that takes up the Eggs. runs that takes up But 10100-70403060 the Eggs more than the other. Propofition 2. In any Series of Numbers in Arithmetical Progreffion, the two Extreams and Number of Terms being given; thence to find the common Difference of all the Terms in that Series. Theorem 2. The Difference betwixt the two Extreams, being S divided by the Number of Terms less than an Unit or I. The Quotient will be the common Difference of the Series. Per Corol, z. EXAMPLE. EXAMPLE 1. One had Twelve Children that differed alike in all their Ages; the youngest was Nine Years old, the eldeft 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. Then 11) 27,5 (2,5 The common Difference required. And A Debt is to be difcharged at eleven feveral Payments to be made in Arithmetical Progreffion. The firft Payment to be Twelve Pounds Ten Shillings, and the laft to be Sixty-three Pounds. What is the whole Debt, and what muft each Payment be? Per Theorem 1. Find the whole Debt thus : 12,5+63 75,5 The Sum of the Extreams. II The Number of Terms. 755 755 2) 830,5 (415,25=415 5s. The whole Debt. Then, per Theorem 2. find the common Difference of each Payment. Thus 63-12,5=50,5 The Difference of the Extreams. Then 10) 50,5 (5,05=51. 15. The common Difference. Confequently 12. 10+5. 1=17. 1=17. 11 The second Payment. 1. S. 1. S. 1. S. And 17. 11+5. 1=22. 12 The third Payment, &c. EXAMPLE 3. A Man is to travel from London to a certain Place in ten Days, and to go but two Miles the firft Day, increafing every Day's Journey by an equal Excefs; fo that the laft Day's Journey may be Twenty-nine Miles; what will each Day's Journey be, and how many Miles is the Place he goes to diftant from London ? First First 29-227 The Difference of the Extreams. And 10-1=9 Then 9) 27 (3. The common Difference. 2) 310 (155 The Distance required. There are eighteen Theorems more relating to Questions in Arithmetical Progreffion; but because they would require a great many Words to fhew the Reafon 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 Proportion continued; fometimes called Geometrical Progreffion. WHE HEN a Rank or Series of Numbers do either increase by one common Multiplicator, or decrease by one common Divifor ; thofe Numbers are faid to be in Geometrical Proportion continued. As 2 . 4 8 16 16 8 32. &c. here 2 is the common Multiplier. 4. &c. here 2 is the common Divifor. 2 6. 18.54. 162. &c. here 3 is the common Multiplier. { 6+ Or{162.54 18.6 2 here 3 is the common Divifor. Note, The common Multiplier (or Divifor) 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 Refpect of two Magnitudes (confequently two Numbers) of the fame 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 lefs than three Terms to form a Proportionality or Similitude of Ratio's; and if but three Terms, the fecond muft fupply the Place of two, As in thefe 2. 4.8. That is, 24: 4: 8. (of:: fee page 5.) Here 4 the middle Term fupplies the Place of two Terms, to wit, of the fecond and third; 8 bearing the fame Reason, Likeness, |