EQU
t 334- ]
feveral arife from End. Ax. 19. Prop. 4. Book 6. and Prop. 47. Lib. I. Elem.
To facilitate this Difcovcry, of the Relations of the Lines in the Figure, there are feveral Things that contri- bute ; as firft, the Addition and Subltraftion of Lines; lir.ee from the Values of the Parts, you may find the Values of the whole; or from the Value of the whule, and one of the Parts, you may obtain the Value of the other Part. Secondly, by the Proportionality of Lines ; fince, as above fuppofed, the Rectangle of the mean Terms, divided by either of the Extremes, gives the Value of the other ; or, which is the fame Thing, if the Values of all four of the Proportionals are firil had, we make an Equality (or jE.qtlP.tion) between the Rectangles of the Extremes and Means. But the Proportionality of Lines is bell found out by the Similiarity of Triangles ; which, as it is known by the Equality of their Angles, the Analyft ought in parti- cular to be converfant in. In Order to which, 'twill be necefiary he be Mafter of Euclid, Prop. 5, 15, 15, 29, and 32, Lib. I. and of Prop. 4, 5, 6, 7, S, Lib. VI. and of rhe 20, ar, 22, 27, and 31, Lib. III. To which may be added, the 3d Prop. Lib. VI. or the 35th and 3 7 rh Prop. Lib. 111. Thirdly, the Calculus is promoted by the Addition, or Subtraction of Squares, viz. in right angled Triangles, we add the Squares of rhe lefTer Sides, to obtain the Square of the greater ; or from the Square of the greater Side, we fubfrracc the Square of one of the lefler, to obtain the Square of the other. On which few Foundations, if we add to them Prop. 1. of the Vlth Elem. when the Bufinefs relates to Superficies, and alfo fomc Propofitions raken out of the nth and 12th of Euclid, when Solids come in Queftion ; the whole analytic Art, as to right-lined Geo- metry, depends. Indeed, ail the Difficulties of Problems may be reduced to the fole Compofition of Lines out of Parts, and the Similiarity of Triangles ; fo rhat there is no Oc- cafion to make Ufe of other Theorems; becaufe they may all be refolved into thefe two, and confequently into the Solutions that may be drawn trom them.
6°. To accommodate thefe Theorems to the Solution of Problems, the Schemes are oft times to be farther con- flrufted, by producing out fomc of the Lines, till they cut others, or become of an affigned Length ; or by drawing Lines parallel, or perpendicular from fome remarkable Point ; or by conjoyning fome remarkable Points ; as alfo, fometimes, by conftrufting them after other Methods, ac- cording as the State of the Problems, and the Theorems, which are made ufe of to folve it, fhall require.
As for Example : If two Lines that do not meet each other, make given Angles, with a certain third Line ; per- haps we produce them fo, that when they concur, or meet, they fhall form a Triangle, whofe Angles, and con- fequently the Ratio of their Sides, fhall be given ; or if any Angle is given, or be equal to any one, we often com- plete it into a Triangle given in Specie, or fimilar to fome other, and that by producing fome of the Lines in the Scheme, or by drawing a Line fubtending an Angle. If the Triangle be an oblique-angled one, we often refolve it into two right-angled ones, by letting fall a Perpendicular. If the Buiinefs concern multilateral, or many fided Figures, we refolve them into Triangles, by drawing diagonal Lines, and fo in others ; always aiming at this End, viz. that the Scheme may be refolved either into given, or fimilar, or righr angled Triangles.
Thus, in the Example propofed, draw the Diagonal B D, that the Trapezium A B C D, may be refolved into the two Triangles, A B D a right angled one, and B D C an oblique-angled one, [Figure 8.) Then refolve the ob- lique-angled one into two right-angled Triangles, by letting fall a Perpendicular from any of its Angle B C or D, upon the oppofitc Side ; as from B upon C D, pro- duced to E, that B E may meet it perpendicularly. Bur lince rhe Angles BAD, and B C D, make in the mean while two right ones, (by 22 Prop. 3 Elem.) as well as B C E and B C D, the Angles BAD, and B C E are perceived to be equal ; confequently the Triangles B C E, and DAB to be fimilar. And fo the Computation (by affirming AD, A B, and B C, as if C D were fought) may be thus carried on, viz. A D and A B, (by Reafon of the right-angled Triangle A B D) give you B D. AD, AB, BD and B C, (by Reafon of the fimilar Triangles A B D, and C E B) give B E, and C E. B D, and B E, (by Reafon of the right-angled BED) give E D : and E D — E C gives C D. Whence there will be obtain'd an Equation between the Value of C D fo found out, and the fmall Algebraic Letter that denotes it. We may alfo (and for the greateft Part it is better fo to do, than to follow the Work too far in one continued Scries) be- gin the Computation from different Principles, or at leaft promote it by divers Methods to one and the fame Conclufion ; that at length there may be obtained two Values of any the fame Quantity ; which may be made equal to one another, Thus, AD, A B, and £ C, give
EQU
B D, BE, and C E as before ; then CD + CP o!v M ED; andlaftly, DB and E D (by Reafon of the riaht angled Triangle BED) give B E. s
7°. Having concerted your Method of Procedure and drawn up your Scheme ; give Names to the Quantities that enter the Computation, (that is, from which af- firmed the Values of others are to be derived, till y ou come to an Equation) chufing fuch as involve all the Conditions of the Problem, and feem accommodated be- fore others ro the Bufincfs, and that fhall render the Conclufion (as far as you can guefs) more fimple, but yer not more than what mall be fufficient for your Pur- pofe : Wherefore, don't give new Names to Quantities, which may be denominated from Names already given. Thus, of a whole Line given, and its Parts, of" the three Sides of a right-angled Triangle, and of three or four Proportionals, fome one of the leaft confidcrable wa leave wirhout a Name ; becaufe its Value may be deriv'd from rhe Names of the reft. As in the Example already- brought, if I make AD = i, and A B = a, I denote B D by no Letter, becaufe ir is the third Side of a right- an gied Trian gle A B D, and confequently its Value is V Ji-aa. Then if I fay, B C = b, fince the Triangles D A B, and B C E are fimilar, and thence the Lines AD, AB:: BC, C E proportional, to three whereof, viz. to A D, A B, and B C, there are already Names given ; for rhat Reafon I leave the fourth C E without a Name, and in its Room I make ufe of — difcover'd from
the foregoing Proportionality. And fo if D C be called' c, I give no name to D E, becaufe from its parts D C, and
C E, or c and — , its Value c 4- — comes out.
X ' X
8. By this Time, the Problem is almolt reduced to an Equation. For after the aforefaid Letters are fet down for the Species of the principal Lines, there remains no- rhing elfe to be done, but that out of thofe Species, the Values of other Lines be made out, according to a pre- conceived Method ; till after fome forefeen Way they come to an Equation. And there is nothing wanting in this Cafe, except that by Means of the righr-angled Triangles B C E and B D E, I can bring out" a double Value of
BS,«. BCq-CEq for*-'"- aabb ^
2 a b c
or xx — a a — cc —
,L?±li\ = BE<1 . x x I J -
q — C E q I or b b -
as alfo B D q — D E q (0
a a b b\ 1
~ X x ) = B E q. And hence I blotting out on both
Sides j you /hall have the Equation b b s= to^t
— a a — cc — ; which bein^ reduced, becomes .v* =:
x x 3 ^ '
+ aa
-j- b b x -}- 2 a b c. as before.
9°. For the Geometry of Curve Lines : We ufe to de- note them, either by defcribing rhem by the local Motion of right Lines, or by ufing Equations indefinitely expref- fing the Relation of right Lines difpofed in Order, ac- cording to fome certain Law, and ending at the Curve Lines. See Curve.
The Antients did the fame by the Seflions of Solids, but lefs commodioufly. The Computations, which regard Curves, defcribed after the firft Manner, are performed as above directed : Thus, fuppofe A K C (Fig. 9.) a Curve Line, defcribed by K, the vertical Point of the Square AK|, whereof one Leg A K, freely Aides rhro* the Point A given in Pofition, while the other K p of a determinate Length is carried along the right Line A D, alfo given in Pofition ; and it is required to find the Point C, in which any right Line C D, given alfo in Pofition, fhall cut this Curve: Draw the right Lines AC, C F, which may reprefent the Square in the Pofi- tion fought, and the Relation of the Lines (without any Difference, or Regard, of what is given or fought, or any Rcfpeft had to the Curve) being confidered ; you perceive rhe Dependency of the others upon C F, and any of thefe four, viz. BC, B F, A F, and A C, to be fynthe- tical ; two whereof you affume, as CF = i, and C B = x ; and Beginning the Computation from thence, you prefently obtain B F = V a "a"— x x, and A B =
. *' - , by Reafon of the right Angle C B F ; and
v a Si ~~ 2G X
that the Lines B F, B C : : B C, A B are continual Pro- portionals. Moreover, from the given Pofition of C D, A D is given, which therefore call b ; there is alfo given the Ratio of B C to B D, which fuppofe as d to e, and
you have B D = ~, and A B = b — -^
an Equation which (by iquaring
e x
Therefore b — ~T -
V:
