EQU
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EQU
jrtfy call it, in the Algebraical Language. The Conditions, thus tranilated to Algebraic Terms, will give as many Equations as are nece&xv to folve it. To illuftratc this by an Inftance : Suppofc'it required to find three Num- bers, in continual Proportion, whofe Sum is 20, and the Sum of the Squares 140 : putting x, y, z, for the Karnes of the three Numbers fought, the Queftion will be tran- slated out of the Verbal l to the Symbolical Expreiuon, thus :
In Symbols.
x, y, z?
.v ; y : : y : z, or
x -f- y -f- z = x x ~r y y ~h z 2
However, to render the Solution of fucli Problems a little more ealy and familiar, we ihall add an Example or two thereof.
i°. Given, the Sum of Pim Numbers a, and the 'Diffe- rence of their Squares b 5 to find the Numbers themfelves. Suppofe the leffer, x 5 the other will be. a — #• and their Squares xx and aa — z&x -- xx: Whofe Difference, Consequently, a a — % a x = b!
. a a — b
a a — b = z a x. Or ■
=yy
- I40.
bring the Queftion into
Symbolically. x y y, 2LY_ ?
y -
-yy.
yy
. _*- = 140.
"The SHefion in Words.
Required three Numbers, on thefe Conditions.
That they be continually pro- portional.
That the Sum be 20.
AndtheSumof theirSquares 140.
Thus, is the Queftion brought to thefe Equations, viz.
- B 3= y y, x -f- 2 -f- y = 20, and A?.a?-|-yy--f-zz:= 140,
by the Help whereof, x, y, and z, are to be found.
The Solutions of QucfHons are, for the moft Part, fo much the more expedite and artificial, by how much the unknown Quantities, you have at firft, are the fewer. Thus, in the Queftion propofed, putting x for the firft Number, and y
for the fecond, — will be the third Proportional j which
being put for the third Number, Equations as follows.
The Queftion in Words*
There are fought three Numbers
in continual Proportion. - - -
Whofe Sum is 20. - - - «
And the Sum of their Squares 140. --------
Xou have therefore the Equations x -J- y -4- ^ = 20,
and x x -\~ y y -I- -2L = 140, by the Reduction whereof,
XX
x and y are to be determined.
Take another Example: A Merchant increases his Eftate annually by a third Part, abating icoZ. which he fpends yearly in his Family j and after three Years he finds his Eftate doubled, ghiery, What is he worth ? To re- folvc this, it muft be obferved, that there are (or lie hid) feveral Proportions, which are all thus found out and laid down.
In Engtifi.
A Merchant has ai Eftate. — — ■ Out of which the firft Year he expends 100 1. And augments the reft by one Third. - And the fecond Year he expends 100/. > — And augments the reft by one Third. - And fo the third Year he expends 100/. And by the reft gains likewiie one Third. And he becomes at length twice as rich as at firft.
Algebraically.
— 2 a a? is call'd b. Whence, by Redu&ioa
' 1 b ._ ( = — a ■ )—x.
a 2 a
E.gr. Suppofe the Sum of the Numbers, or a, to be S, and the Difference of their Squares, orb, 16: Then will,
— a— — (—4—1) ss= 3 9sx. And a — x — 5. There- 2 3 a v J
fore the Numbers arc 3 and 5.
2 . To find three Quantities x, y, and z; the Sum of each 'Pair ivhereof is given. Suppofe the Sum of the Pair x and y be a; that of x and z, b: and that of y and z, c. To determine the three Numbers required, x, y, and z; we have three Equations # -- y = a j x -f- z = b $ and y -j- z = c : Now, to exterminate two of the unknown Quantities, e.gr. y and x 5 take away x, both from the firft mid {econd Equations ; and we fhall have y~ a — x and z = b — x. Which Values being fubftituted for y and , z in the third Equation 5 there will arife a — x ~\- b — x
= c 1 and by Reduction x ■= — # Having found x
2 the former Equations, y = a — x and z = b — x will give y and z.
Thus, e. gr. If the Sum of the Pair x and y be 9 $ of x and z, ro ; and of y and z, 13 : Then, in the Values x, y . and z, write 9 for a, 10 for b, and 13 for c; and you wilj
havea-j- b — c = tf- and consequently x ( = --I- 1^
s= 3, y (= a — x") =6 and z (= b — x~) =s 7.
3 . To divide a given ghtantity into any Ntrmber of Tarts, fo as that the greater 'Pans pall exceed the leajl by any given Differences. Suppofe A a Quantity to be divided into four fuch Pares, the firft and fmalleft whereof is x ; the Excefs of the fecond Part above this, b, of the third, c, and the fourth, d : Then will x -f- b be the fecond Part, x ~\- c the third, and x -j- d the fourth : The Aggregate of all which 437 -J- b-|-c -j- d is equal to the whole Line a. Now, taking away from each, b -/- c -f- d,
and there remains 4 x = a — b — c — dor x = •
4
Suppofe, e. gr. a Line of 20 Feet,, to be divided into 4
Parts, in fuch manner, as that the Excefs of the fecond
above the firft may be % Feet, of the third, 3 Feet, and
of the fourth 7 Feet. Then the four Parts will be x
(=
a — b — c — d
\6x — 3700 i6x — 37°°
— 9 +~T 7 >'
6 4 x — ■ 14800
=7
Therefore the Queftion is hrought to this Equation,
— = * x, by the Reduction whereof you will find
x = 14800.
,r. Viz. Multiply it into 17, and you have £4.1?— 14S00 = 54#5 fubtracf 54^, and there remains ro* — ^Soo— o,or 10 #= 148CO; and dividing by 10, you have x — 1480. So that the Value of his Eftate at firft was 1480 Lib.
It appears then, that to the Solution of Queftions about Kumbers, or the Relations of ab.ftracT: Quantities; there is fcarcc any Thing more required, but to tranfiate 'em cut of the common, into Algebraic Language - 7 i. e. into Characters, proper to exprefs our Idea's of the Relations of Quantities. Indeed, it may fometimes "happen, that the Language whetein the Queftion is ftated, may feem unfit to be rendted into the Algebraical ; tho' by making a few Alterations therein, and attending to the Senfe, rather than the Sound of the Words, the Tranflation becomes cafy enough. The Difficulty here refults merely from the Difference of Idioms, which is as obfervable between moft Languages, as between the common and fymbolical.
- or ^iTJ^lT.) = t, x + h = 4,
x -\- c = 5, and x -\- d=c>. And after the fame Man- ner, may a Quantity be divided into a greater Number of Parts on the fame Conditions.
4°. A 'Perfou diffofed to distribute a little Money among fome Beggars ; wants Eight 'Pence, of three Vence for each of 'em : He therefore gives 'em Thvo 'Pence apiece, and has three Pence left: the Number of Beggars is required. Let the Number of Beggars be call'd x ; and the Perfon want Eight Pence of giving 'em all 3 a;. Confe- quently he has 3 x — 8 ; out of which he gives 2 x Pence 5 and the remaining Pence are 3. That is a? — 8 = 3 or »== 11.
5 . the Power or Strength of one Agent being given- to determine hoiv many fuch Agents will produce a given EffeiJ a, in a given time b. Suppofe the Power of the Agent fuch, as that it may produce the EftecT: c, in the Time d 5 then, as the Time d, is to the Time b, fo is the Effect c, which the Agent can produce in the Time d, to the Effect it can produce in the Time b, which accordingly
will be -— Then, as the Effect: of one Agent — is to the
d d
joynt Effect: of 'cm all a ; fo is that one Agent, to all the Agents. Confecmently the Number of Agents wiil be a d be.
Thus, e. gr. If a Clerk, or Writer, in 8 Days Time, tranferibe 15 Sheets; how many fuch Clerks are required 9 ~
to tranferibe 405 Sheets in 9 Days ? Anf. 24. For if 8 be fubftituted for d, 15 for c, 405 for a, and p for b, the
That is 1312' or , 4 . MS
ad ... , 4°5X>
Number g^ will become -jTli
5°. the Powers of feveral Agents being given; to de- termine the time x, wherein they will joyntly perforin a given Effcft d. Suppofe the Powers of the Agents, Oi, C, fuch as that in the Times e, f, g, they would pro- duce
