Ix Sect. 2. gultiplication of Surd Quantities. Gafe 1. WHEN : Quantities are pure deaths of the fame Kind; multiply them together, and to their Product prefix their radical Sign. EXAMPLES. Waa -b b 31v bald beaat deaalvaaaa -bbbb Cafe 2. If Surd Quantities of the fame Kind (as before) are joined to rational Quantities, then multiply the rational into the sational; and the Surd into the Surd, and join their Products together. EXAMPLES da bo 15od & bat da + 15 V ab 2/3 bra 3 avca Ix21313db v bcal 15 cdav bcaa + dc, a 175 vabd Sect. 6. Diviliori of Surd Quanlilies. Cafe 1. WHEN the Quantities are pure Surds of the fame Kind, , and can be divided off, (viz. without leaving a Remainder) divide them, and to their Quotient prefix their radical Sign. EXAMPLES. V aa-bb 1-2131v a lvbatda Vaa+bb Cafe 2. If Surd Quantities, of the same Kind, are joined to rational Quantities; then divide the rational by the rational, if ir can be, and to their Quotient join the Quotient of the Surd divided by the Surd with it's firft radical Sign. E X A M P L E S. 23bva 3avca 15 vab Note, 1 Note, If any Square be divided by it's Root, the Quotient will be it's Root. E X A M P L E S. bb+2bctrcfaaa a— 2 bba a +bbbb 2v alv0b+2bctrolu at -26baa +6+ 1:-23 V alv 6b+2bc toolv 24 - 2bbaa +64 Hvalve Sect. 4. Involution of Surd Quantities. they are involved to the same Height as their Index denotes, by only taking away their radical Sign. EXAMPLES Case 2. When the Surds are joined to rational Quantities ; involve the rational Quantities to the fame Height as the Index of the Surd denotes; then multiply those involved Quantities into the Surd Quantities, after their radical Sign is taken away, as before. E X AMPLES. Il BV al 50 V cal36 V aa dd 21bba 25 ddea lobbaa9bbdd 12 14:3V bol 3d: 5v aa +6b da: 3Vb abc 27 dddaa +27 dddbolddda a ab I The Reason of only taking away the radical Sign, as in Cafe 1. is easily conceived, if you consider that any Root being involved into itrelf, produces a Square, &c. And from thence the Reason of those Operations performed by the second Case may be thus stated. Supposeb v a=x. Then Ve=% per Axiom 4. and both Sides of the Equation being equally involved, it will be a = Then multiplying both sides of the Equation into bb, iç becomes b ba=** per Axium 3. Which was to be proved. Again, bb Again, Let s dvca=*: Then vra= 5 and ca 25 dd Suppose { Allo from hence it will be easy to deduce the Reason of multiplying Surd Quantities, according to both the Cases. For IV bax } Example 1. Cafe 1. =x* 61v ba=2*. which was to be proved. I Q2 2 @ Let { 236V }Example 1. Cafea 1:03 V bc= 1 2:30 4 Var 3b 4*35 V abo= 36d from what is proved above. 5x36 dl 6 136 d v bca=2*, &c. for the rest. Division being the Converse to Multiplication, needs no other Proof, CH A P. V. Concerning the Nature of Equations and how to prepare them for a Solution. WHEN any Problem or Question is proposed to be analy tically resolved; it is very requisite that the true Design or Meaning thereof, be fully and clearly comprehended (in all it's Parts) that so it may be truly abstracted from such ambiguous Words as Questions of this Kind are often disguised with; ocherwise it will be very difficult, if not imposible, to state the Question right in it's fubftituted Letters, and ever to bring it to an Equation by such various Methods of ordering those Letters as the Nature of the Questions may require. Now Now the Knowledge of this difficult Part of the Work is only to be obtained by Practice, and a careful minding the Solution of such Jeading Questions as are in themselves very easy. And for that keason I have inserted a Collection of several Questions; wberein there is great Variety. Having got to clear an Understanding of the Question proposed, as to place down all the Quantities concerned in their due Order, viz. all the substituted Letters, in such Order as their Nature requires; the next Thing must be to consider whether it be limited or not. That is, whether it admits of more Anfwers than one. And to discover that, observe the two following Rules. RULE 1. When the Number of the Quantities fought exceed the Number of the given Equations, the Question is capable of innumerable Answers. E X A M P L E. be 22. Suppose a Question were proposed thus, there are three such Numbers, that if the first be added to the second, their Sum will And if the second be added to the third, their Sum will be 46. What are those Numbers ? Let the three Numbers be represented by three Letters, thus, call the first a, the second e, and the third y. Then ate=22 { 75346 } according to the Question. Here the Number of Quantities fought are three ; a, e, y, and the Number of the given Equations are but two. Therefore this Question is not limited, but admits of various Answers; because for any one of thole three Letters you may take any Number at Pleasure, that is lets than 22. Which with a little Confideration will be very easy to conceive. RULE 2. When the Number of the given Equations (not depending upon one another) are just as many as the Number of the Quantities fought; then is the Question truly limited, viz. each Quantity fought hati but one single Value. As for instance, let the aforesaid Question be proposed thus, There are three Numbers (a, e, and y, as before) if the first be added to the second, their Sum will be 22; if the second be added to to the third, their Sum will be 46; and if the first be added to the third, their Sum will be 36. What are the Numbers ? That is, a te=22. e+y = 46. and a +y = 36. Now the Question is perfectly limited, each single Quantity having but one fingle Value, to wit a = 6, e=16, and y = 30. ' N. B. If the Number of the given Equations exceeds the Number of the Quantities fought; they not only limit the Question, but oftentimes render it impossible, by being proposed inconsistent one to another. Having truly stated the Question in it's substituted Letters, and found it limited to one Answer (or at least so bounded as to have a certain determinate Number of Answers) then let all those sube ficuted Letters be so ordered or compared together, either by adding, subtracting, multiplying, or dividing them, &c. according as the Nature of the Question requires, until all the unknown Quantities except one, are cast off or vanished ; but therein great Care muft be taken to keep them to an exact Equality ; and when that unknown Quantity, or some Power of it (as Square, Cube, &c.) is found equal to those that are known; then the Question is said to be brought to an Equation, and consequently to a Solution, viz. fitted for an Answer. But no particular Rules can be prescribed for the casting off, or getting away Quantities out of an Equation ; that Part of the Art is only to be obtained by Care and Practice. And when that is done, ić generally happens so, that the unknown Quantity which is retained in the Equation, is so mixed and entangled with those that are known; that it often requires some Trouble and Skill to bring it (or it's Powers, &c.) to one side of the Equation, and those that are known to the other Side; (ftill keeping them to a juft Equality) which the ingenious Mr Scooten in his Prin, tipia Matheseos Universalis, calls Reduction of Equations. The Business of reducing Equations (as of most, if not all Algebraick Operations) is grounded and depends upon a right Application of the five Axioms proposed in Page 146, and therefore, if those Axioms be well understood, the Reason of fuch Operations must needs appear very plain, and the Work be easily performed; as in the following Sections. |