304/a2+26+7=400-40e†ee 4—ee 5a2+e2+=400—40% 2,5 6140=400—40e 610 714=40—4e. From 846=40-14=26 26 8÷49e= 6 to be fubftituted in the first Step, &t. 4 4lw2 5 ea = +3 4. 11 X 4 a2+6a2+9 9 93a3-3a= 3a3-3a =5aa 102 11 9aa+18a*+9aa 4a2+24a2+3° 9a0—18a+ +9a2 I 12xa*+6a2+9|13|9a6-18a9a1=5a°+42a*+117a2+108 13—5a6—42a3—117a2 14 4a0-60a4-108a2=108 14415 a 15aa—27a1—27 Subftitute 16a2=y Then 17 y3 — 15y3 — 273=27 16 18 ay +3 4 5aa +3 4 =5a2+12 (In, 115.) Example Sub. for a in the 256e+ ft Step divi-13 2—12y3e2-845e+158e-9472e=0 dingby 256eeee 13÷21434—6y3e2-425e79e8-4736e=Q 14+6y2+ 15 3*+79e2-4736e*=6y3e +4zje 42ye бyyee+42e 1780e 1920s-1496e^+9216e2+23040614400 =y 15÷6y3e2 163-796-4737* &c. Sub. for y 16 48e+5766+720e2 16 36° +36e4502 17÷ 185e+12es—281e*+576e2+1440e +900 =y 18 19 256 +120e13—2810e2 +5904e1 +7656e +87961c +13824e7 ·289152e6—787680c3—174024e+1658880e3+31104002 2592000+810000÷ge" +216 +270e +1296e +3240es +2025ea=3y=c+96e +120. R •19 21÷16 20 2562 + 12013- 2810e25904€2 + 7656€ + 87961 +13824e-28915266787680c-1740246+ +1658880e2+311040062 +25920006+810000= 9e" +1080e" +1350 +220320" + 55080€ + 3442568 +124416c+ 466560€ + 583200es + 243000e4. 21 161696061-4160e1-16128e-474246 + 13708802s 5353668 110592e7 +810000=0. 75571266 -417024e4+1658880e3+3110400e2+2592000€ 226-60e-2601008102964e9+ 3346ts. 6912e7-472326-856806-26064e+ +103680e3 + 19440062+162000e+50625=0. An Equation of fixteen Dimensions, whofe Root e will be found to be 5 (În. 509.) whence a=e*+96e+1201‡e 2=3 (Step. 7th.) 4e SCHOLIUM XIX. 530. Hither may be referred Sir Ifaac Newton's Rules for the Extermination of unknown Quantities-in adfected Equations. Vid. Newton's Algebra, p. 65, or in the best Latin Edition, p. 73. СНАР. XI. Of bringing Questions into Equations. PARTITION VII. 531. Algebraical Questions art of btw Sorts, Determinate and Indeterminate, according to the Number of Equations and unknown Terms whereof they confift. DEFINITION XXIX. 532. A Determinate Question is that which confifts of as many independent Equations, as unknown Quantities: As if it were required to find what two Numbers those are, a and e. whereof ac =24 and 2 a = 6. Or to find what three Numbers thofe are, whereof ate=25, a+y=28, e+y= 31. Which kind of Queftions are called Determinate, because they admit of no more Anfwers than the Number of the Dimensions of the last Quantity fought. DEFINITION DEFINITION XXX. 533. An Indeterminate Queftion is that which confifts of more unknown Quantities than Equations: As if it were required to find what two Numbers thofe are, a and e, whereof ae 24: Or to find what three Numbers thofe are, whereof ate=25, and aty 28: Which Sort of Queftions are called Indeterminate, because they admit each of innumerable Answers. Ex. gr. Any two Numbers whofe Product is 24 will answer the former, and in answering the latter, any Number lefs than 25 may be affumed for one of the unknown Quantities. SCHOLIUM XX. 534. If in any Queftion the Number of Equations be greater than the Number of unknown Quantities; it is odds but fome of the Equations are contradictory, and confequently the Solution of fuck Queftion is impoffible. PARTITION VIII. 535. Again, Particular. Algebraical Queftions may be divided into General and DEFINITION XXXI. 536. A General Question is one expreffed in general Terms, the Solution of which affords Theorems for particular Cafes; as when it is required to inveftigate a Theorem for Determining any three Numbers or Quantities a, e, y from the Sums of every two of them given, viz. a+e=b, a+y=c, e+y=d, where for b, c, and d, may be affumed any three Numbers at Pleasure. DEFINITION XXXII. 537. A Particular Question is expreffed in particular Terms; as when it is required to find three Numbers a, e, y, on these Conditions that ate=25, ate 28, and e+y=31. PROBLEM XXX. 538. To bring a Queftion to an Equation. Effection. The whole Art of bringing Questions into Equations confists in a due Expreffion of all their Quantities by proper Species, (the known Quantities by. Confonants (or Numbers) and the unknown ones by Vowels (In. 530.) which is to be learned by Example rather than Precept. Example 1. A Lady feeing divers poor Perfons at her Door was willing to diftribute fome Money among them, but when the Number'd them, fhe found that she wanted Six-pence to give Four-pence a piece to them; fhe therefore gave to each Three-pence, and had Two pence remaining: What was the Number of poor People, and what Money had the in her Pocket? A Lay The Questions in Words. A Lady feeing divers poor Perfons at her Door for whofe Numbers put was willing to diftribute fome Money among them, for the Quantity of which put But when fhe Number'd them she wanted Six-pence to give them Four-pence a-piece, i. e. She therefore gave to each Threepence, and had Two-pence remaining, i. e. The fame in Species. 4a=e+6. 3a=e-2 The Queftion then confifts of two Equations, and two unknown Quantities a, e. Whence by (In 527.) a will be found 8 Pence and e=26 Pence or 2 s. 2d. Example 2. One bought a Horse, and Sold it again for eleven Pounds, in the felling it gained as much per. Cent. as the Horfe coft him. What did he give for the Horfe? The Questions in Words. One bought a Horfe whofe Price was And fold it again for 117. or So that his Gain was In the felling of which he gain'd as much by the 100%. or As the Horfe coft him, i. e. The fame in Species. a b b-a C a:b-a-c:a Whence we have this Equation aa=bc—ac (In. 189.) which by due Reduc Example 3. A Father and his Son went to the Wood for each a Burthen of Sticks, and in their return Home the Son complained that he was over-loaded; to whom the Father replied, if I take ten of your Sticks, then shall I have twice the Number of Sticks that you have, and if you take ten of mine we fhall have an equal Number. It is required to find the Number of Sticks that each had. Here if a be put to reprefent the Number of Sticks which the Father had, and e the Number of Sticks which the Son had; and if for 10 be put b, then will the Question be expreffed by these two Equations: 11a+b=2xe=b=2e-2b Whence by due Reduction, it will be found that e=5b50 and a=e+ab F76=72. (In. 527.) Example |