EXAMPLE. Let it be required to find the Decimal Parts equivalent to 17 s. 9 d. 2 Farthings. Firft 0,05=15. Therefore 17x,05,85....=175. Confequently their Sum, viz. 0,889577=175.91d. Now to find the Value of Decimals in known Parts of Coin or Weights, &c. is only the Converfe of the former Work, and is thus performed. Multiply the given Decimals with the Denominator of the Vulgar Fraction required: That is, multiply the Decimals with fuch a Number of Units, as are contained in the next lower Denomination of that Kind or Species which your Decimal is of; and the Product will be the Number required. EXAMPLE. I 1. What is the Value of 0,825 Decimals of 1 Pound Sterling; That is, how many Shillings, Pence, &c.,825? First, the next lower Denomination is 20, because 20 s. make one Pound. Therefore 0,825 20 Shillings 16,500 and Parts of Shilling. 12 6,000 Anfwer 0,825 16 s. 6d. Again, What are the known Parts of English Coin equal to 3,666666 Decimals? Here the 3 Integers are 3 Pounds. Then,666666 20 What is the Value of 0,74722 Parts of 1 lb Troy? First, 74722 Then, ,96664 Again, ,33280 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 confift more in Geometrical Computations, than in the common or practical Parts of Arithmetick, as will appear further useful upon feveral Acon; although even in those they are very counts; 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 proper 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, 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 3. And 1243. Again, 12 x 6. And 12-26, &c. From hence it follows, that if any Number be divided by a proper 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. 4 : 1 :: 12: 48, &c. But the Truth of these will be best 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 HEN 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. As If any three Numbers be in Arithmetical Progreffion, the Sum of the two Extreams (viz. the first and laft) will be equal to the Double of the Mean or middle Number. As in these, 2.4.6. Or 3.6.9. Or 3.7. II Viz. 2+6=4+4. Or3+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 Viz. 2+8=4+6. And 3+12=6+9. &c. Corollary 1. 9 12.. From these two Lemma's it is eafy to conceive, that if ever 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 10 12 14 2+16=4+14=6+12=8+ 10. Then Or if the Number of Terms be odd, as thefe, 2.4.6.8. 10. 12. 14. 16. 18. &c. Then 2+18=4+16=6+14=8+12=10+10. Lemma 3. Each Term in every Series of Numbers in Arithmetical Progreffion is compofed of the Interval or common Difference, fo often repeated, and added to the first, as there are Terms in the Progreffion, after 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 (iz. 1 and 17) is compofed of the common Difference, multiplied into the Number of all the Terms, excepting the firft. 17. As in the aforefaid Progreffion, I. 3. 5. 7. 9. 11. 13. 15. The Number of Terms without the firft is 8 Multiply The common Differences The Difference betwixt the two Extreams Propofition I. 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 Number of all the Terms; and divide the Product by 2. The Quotient will be the Sum of all that Series. Per Corol. I. 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. 12 26 13 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 these 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 Queftion 200+2=202 Is the Sum of the two Extr. 100 Is the Number of all the Terms. The Number of 2) 20200 (10100 Yards he runs that takes up the Eggs. And Then Now 4 Miles 7040 Yards The Yards he runs that takes up But 10100-7040 3060l 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 2.1 The Difference betwixt the two Extreams, being divided by the Number of Terms leffened by Unity ar. 1. the Quotient will be the common Difference if the Series. Per Carol. 2. EXAMPLE. EXAMPLE I. 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. The Number of Terms lefs an Unit. And Then 11) 27,5 (2,5 The common Difference required. Confequently 9+2,5=11,5 The Age of the youngest but one. And 11,5+2,5=14 The Age of the youngest but two. And fo on for the reft. Per Coral. 2. EXAMPLE 2. A Debt is to be discharged 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=5%. 15. The common Difference. 1. S. 1. S. 1. s. Confequently 12.10 +5.1=17. 11 The fecond Payment. 1. S. 7. 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 first 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? L 2 Firft |