EQU
I 333 ]
duce Effefts a, b, c, refpectiyely : And theft, in the Time
x, would produce Effefls *i H — . Confequently 15 e f g e
, d
+ -j- + — = d, and by Reduction * = a b c .
f E e+f+1 .
Suppofc, e. gt. three Workmen cou'd finifh Work, on fuch and fuch Times, viz. A once M
EQU
If the Baft be fought, I p u tAB = c, CD — a? and
?S>° rB ,?.™ b- ? he -?' dra * in S AC, as the Triangles ABC and CBE arc fimilar 5 AB : BC::BC:BE or c<:
b::b:BE. Wherefore B E = ±k. AndCE='|CD
right, CEq +
And fince the Angle CEB b 4
three B E q = B Cq; that is, \ xx ■
bb
Weeks, B thrice in eight Weeks, and C five Times in twelve t j .^ w kj c i, rct J uce d, gives the Bafis fought, x.
an Equtl-
Weeks : And it is required, in what Time they will finifh it together? Here, the Powers of the Agents A, B, C, are fuch as in the Times 3, 8, and 11, refpeilively produce Effects 1, 5, 5; and it is enquired in what Time they will produce the Effect, 1. For a, b, c, d, e, f, g, write 1, 3, 5,
1, 3, 8, 12 ; and there will come out x = ? , I + A or 9
- ' s
of a Week, that is, 6 Days, 5 j of an Hour; the Time they
will finifh it in together.
<S°. Given, the fpecific Gravities of a Mixture, and of Reduction, gives * required. the feveral Ingredients thereof; to find the Proportion of Thus is ' th( , Caladm for the Ingredients therein. Suppofe e the ipecihc Gravity of the Mixture A + B ; a that of A, and b that of B : Since the abfolute Gravity, or Weight, is compounded of the Bulk of the Body, and its fpecific Gravity; a A will be the Weight of A ; B b that ot B ; and e A + e B the Weight of the Aggregate A + B. Confequently, a A + bB = eA-f-eB; and therefore aA — eA=eB — bB ore — b. a — e : : A. B. ibcrtv
Thus, e.gr. fuppofe the fpecific Gravity of Gold to be m.,^iA. as 19, that of Silver as 10 1, and that of K. Hiero's Crown as 1 7 ; then will 10. 3 ( : : c — b. a — e : : A. B) : : the
Lafily, If the Side B C, or B D, were fought, I put A B = c ; C D = a, and BCor BIJ = .f. Then, drawing A C, the Triangles ABC, and C B E, being fimilar ; we have A E : B C : : B C : B E ; or c : x ■. -. x : BE.
Wherefore, BE=H. And C E = \ C D or J a 5 And
the Angle CEB being right CEq + BEq==BCq.
That is I a a -f- i! = * x. An Equation, which by
arriving at the. Equation, as well as the Equation it fclfj the fame in all the Cafes; except that the fame Lines are defign'd by different Let- ters, according as they arc Data, or fpucCjita. Indeed, as the Data, or ghitefita differ, there anils a Difference in the Reduction of the Equation found : But no Difference in the Equation it felf. So that we need make no Dif- ference between given and fought Quantities; but arc at > ftate the Queftion with fuch Data and $U£jlta, as wc think moil favourable to the Solution 01 the Queftion.
A Problem, then, being propofed, compare the
Bulk of Gold in the Crown, to the Bulk o f Silver: Or Quantities it includes; and, without making any Difte-
190. 31 (:: 19 x 10. 10 -f X 3 : : a x e — b. b X a- — e) : : The Weight of the Gold in the Crown, to the Weight of the Silver, and 221. 31 : : the Weight of the Crown to the Weight of Silver.
To bring Geometrical 'Problems to Equations.
Geometrical Qucilions, or thofe relating to continued Quantities, are fometimes brought to Equations, after the fame manner as Arithmetical ones. So that the 1 /? Rule to be here prefcribed is, to obferve every Thing directed for the Solution of Numerical Problems.
Suppofe, e. gr. it were required, to cut a right Line, as A B, (T"ab. Algebra. Fig. <5.) in mean and extreme Pro- bation in C ; that is, fo as that B E, the Squate of the greateft Part, Jhall be Equal to the Reclangle B D, con- tain'd under the whole and the leaft Part.
Here fuppofing A B = a and C B = * ; then will AC = a — x ; and x x ;= a into a — *. An Equation, which by Reduction gives *== — i a -J- ■/ 4a a.
But 'tis very rare, that Geometrical Problems brought to Equation ; as being generally found to depend on various complex Pofitions, and Relations of Lines ; fo that, here, fome further Artifice, and certain fpeciai Rules, will be required, to bring 'em to Algebraic Terms. Indeed, 'tis very difficult to prefcribe any Thing precife in fuch Cafes : Every Man's own Genius fliould he the Rule of his Procedure.
Something, however, fhall be faid in the general, for the
Sake of fucK as ate nothing vcrfed in fuch Operations; and propo-ed that principally from Sir /. Neivton. cZu'rs
Obferve then, 2°. That Problems concerning Lines related ot the . Quantities
rence between Data and Gtiigfita, confider what Dcpcnd- ances they have of each other; that you may learn which of 'em will, by Compofition, give the reft, In the doing of which, it is not neceffary you fhou'd at firft contrive, how fome may be deduced out of others, by an Algebraic Calculus ; It fuffices you temark in the general, that they may be deduced by fome direct Connection.
Thus, e. gr. If the Queflion be about the Diameter of a Circle, A D (Fig. 8.) and three Lines A B, B C, and CD, inferibed in a Semi-circle; whereof, the reft being given, B C is required : It is evident at firft Sight, that the Diameter A D, determines the Semi-circle ; as alfo, that the Lines A B, and C D, by Infcription, determine the Points B and C, and confequently B C required, and that by a direel Connect ion. Yet does not it appear how B C is deduced from the fame Data, by any analytical Cal- culus.
4°. Having confider'd the fevcral Manners, wherein the .i Terms of the Queftion may be cxplain'd and decompound- ed ; chufe fome of the fymhetic Methods ; affuming fome Lines as given, from which there is the moil eafy Acccfs, or Progrefiion to the reft, and to which the Regteffion is the molt difficult. For, tho' the Calculus may be carried on divers Ways, yet muft it begin with thefe Lines. And the Queflion is more readily folved, by fuppofing it to be of thefe Data, and fome gHltefltum readily flowing from 'em ; than by confidering the Queftion as 'tis actually
to each other in any definite manner, ftated, by fuppofing different Quantities, the ghizflta, or Things fought, from different Data: Yet ftill, with what- ever Date ani ghtieftta the Queftion is propofed, its Solution will arife after the very fame manner, without the leafl Alteration of any Circumftancc, except in the imaginary Species of Lines, or the Names whereby the Data are diftin- guifh'd from the ghiiefita.
Suppofc, e.gr. the Queftion were about an Ifofceles Tri- angle, BCD, (Fig. 7.) inferibed in a Circle; whofe Sides, BC, BD, and Bafe CD, were to be compared with the Diameter of the Circle A B. Here, the Queftion may be either propofed, of the inveftigating the Diameter, from the given Sides and Baft ; or of inveftigating the Bafe, from the Sides and Diameter given. Or, lafily, of finding the Sides, from the Bafe and Diameter given : And propofe it under
the Example already given, if from the reft- tides given, it were required to find AD; may be"varioufly P erceivin g tnat this cannot be done fynthetically ; yet that the @Unelita or ^ lt were ^° done ' ' cou '^ proceed in my Ratiocination on it in a direct. Connection from one Thing to others ; I affume A D as given, and begin to compute as if it was given indeed, and fome of the other Quantities, viz. fome of the given ones, as AB, BC, or CD, were fought. And thus, by carrying on the Computation, from the Quantities affumed to the others, as the Relations of the Lines to one another direct, there will always be obtained an Equa- tion between two Values of fome one Quantity; whether one of thofe Values be a Letter fet down as a Reprefen- tation, or Name, at the Beginning of the Work for that Quantity ; and the other, a Value of it found out by Computation: or whether both be found by Computation made different Ways.
5°. Having thus compar'd the Terms of the Queftion
auired,
... u v in general, there is further Thought and Addrcfs requ which Form you will, it will be brought to an Equation, oy g , » » o . . -i- ,
to find the particular Connections, or Relations ot the Lines, fit for Computation. For, what to a Pcrfon who
the fame Algebraic Series.
Thus, if\hc Diameter be fought, I putAB = *, CD
==a, andBC, orBD = b. Triangles ABC and CBE
BC : BE, Or** : b :: b
and C E = \ CD or \ a.
Then, drawing A C, are fimilar; AB :
the
BC
bb
does not fo throughly confider them, may feem imme- diately, and by a very near Relation, connected together j when we come to exprefs that Relation Algebraically, arej often found to require a longer Circuir; and fliall even obligei you to begin your Schemes a-new, and carry on youir Computation Step by Step ; as may appear by finding, B C, from AD, A B, and C D. For you arc only to = b b. Which Equation being reduced, gives proceed by fuch Propofitions, or Enunciations, as can be
fitly rcprcfented in Algebraic Terms, whereof there arc Xx * f cve '.
B E. Wherefore B E = - And, by reafon the Angle
CEB is a right Angle, C E q + B E q = B C q, that is, iaa+V
K
the Diameter required
