Question 15. When z and x are given, to find the reft. Thefe fifteen Queftions are propofed in Dr Pell's Algebra; but he purfues only the firft Question throughout, and breaks off in the other fourteen, after the Values of what I call & and are found. But I have proceeded in every one of them, to to find the Values of all the unknown Quantities, because they afford fuch Variety, as being well obferved by a Learner, will be found very useful in the Solution of moft Questions. Note, I have chose to use the fame Numbers for the respective Value of each Quantity throughout all the Questions, because they will be more fatisfactory in proving the Work than various Numbers would have been. Not but that any Numbers may be taken at Pleafure, provided that the Number reprefented by 4, be greater than that by e, &c. I have omitted the Numerical Calculations purely for the Learner to practise on. Question 16. There are two Numbers, the Sum of their Squares is 2368; and the greater of them is in Proportion to the lefs, What are these Numbers? as 6 to 1. Let a= the greater Number, the leffer, and z=2368. Then i aa+ee=% by the Question. And 2a:e: : 6: : 1 2. 3 10=6e 32 4aa36 ee 45ee=2- -36ee 5+36ee 6 37 e ex Question 17. There are three Numbers in continued Proportion, the Sum of the Extreams is 156, and the Mean is 72; What are the two Extreams? That is, Suppofle a . m e in, and m=72. Then{ 3 x 4 51 4 al = 4 m m 45 4 ·56002ae+ee=55 — 4 m m Question 18. There are three Numbers in, their Sum is 74, and the Sum of their Squares is 1924; What are those Numbers? That is, a, e, y are in ÷ Then zaa+ee+yy=z=1924 Quære a, e, y. 6+7 2 zayzee 8aa2ay+yy=z+ee 8 and 9 10%+ee=ss-asetee 10 +11 2 se=ss - %% Note, In all Questions about continual Proportionals, (either Arithmetical or Geometrical) where three Terms are fought, the Mean is the cafieft found firft (as above) and if all the Terms be Affirmative, then it is equal whether the firft or laft Term be the greatest. 1 Question their Sum is Question 19. There are three Numbers in 76; and if the Sum of the Extreams be multiplied into the Mean, that Product will be 1248; What are thofe Numbers? 6ee=se-p gee-se=-1 8ee-set 455 = 455-p 9+$510e=is+✔äss-p= {52. per Theorem 3. 24 Chap. 8. 14 ພ 15a-y=√ 400 = 20 11 + 15 16 20 = 52 +20=72 16÷217a=36 {and=36 N. B. If you take es+✔ss-p=52 (at the 10th Step) then it will be 76-5224a+y, which is impoffible, viz. that the Mean fhould be greater than the Sum of the two Extreams. Therefore it must be e-is-√15. (See page 201.) с -p=24. Question 20. There are three Numbers in Arithmetical Progreffion, the first being added to twice the fecond, and three times the third, their Sum will be 62; and the Sum of all their Squares is 275; What are thofe Numbers? Suppofe a, e, y in Arithmetical Progreffion. And { 2a +24+3y=62 by the Question. Then 4a+y=ze, per Sect. 1. Chap. 6. 6e+y=31-e 7y=312 e 802 8. 2 9|4a=16ee248e+961 9+1011 aa+yy=20ee372ef1927 3 13- 372 e 1 4 2 1 ee - 372 e 1647 142115ee- 16 2 17 e- • +9, or 8 & the Mean. 36-315, or 3431=3 31 18 13, or 31-17=13 Question 21. There are three Numbers in Arithmetical Progreffion; the Square of the firft Term being added to the Product of the other two is 576; the Square of the Mean being added to the Product of the two Extreams, make 612; and the Square of the laft Term being added to the Product of the firft into the fecond, is 792: What are thofe Numbers? Suppofe a, e, y in Arith. Progref. as before. Then '.' }} 3ee+ya=612 by the Queftion. 15a+y=2 e, per Sect. 1. Chap. 6. 2 +4 7aa+ye+yy+ae=1368 8aa+yy=1368 -2ee 9 x 210 2ya 1224. -2ee 81011 aa+2ya+yy=25924ee 2 5 12 aa+zya+yy=4ee 11 and 12134ee=2592 13+4ee148ee = 2592 14815ee = 324 4 вв 15 w2 16 = 324 = 18, the Mean. 17-18, 19! aa—zya+yy720 576144 |