Q.UA
The moft eminent of thefe Quadratrices are that of 2), voftrates and that of Mr. Wjbitirnbaufen for the Circle, and that of Mr. 2>erfo for the Hyperbola.
Quadratrix of 2)inoJh r ates, is a Curve, A M#?#*, (Tab. Analysis, Fig. zz. J whereby the Quadrature of the Circle is effected, tho 1 not Geometrically, but Mecha- nically 5 thus call'd from its Inventor '2)inoJlrates.
ltsGcnefisis thus, Divide the Quadrant al Arch ANB
into any Number of equal Parts; in N n, &c. by a conti- nual Biflcclion divide the Radius A C into the fame
Number of Parts in the Points P^, &c. Draw Radii C N, en, &c. Laftly, on the Points P p 3 c5c. erect Perpen- diculars P M, p ffi, &c. the Curve form'd by connecting thefe Lines is the Quadratrix of 2)inofirate$.
Here, from the Conftruftion, A B : A N : : A C =APj and therefore, if AB = fl, AC=£, AN = w, AP==y J ax=-by. See Quadrature..
Quadratrix ^tfchimhaufianajs atranfcendental Curve AM«z?#B, (Fig.x^.~) whereby the Quadrature of the Circle is likewife effected 5 invented by M. Tfchimhaufen, in imitation cf that ofjDinoJlrates.
Its Genefis is thus conceived— —Divide the Quadrant ANB, and its Radius AC into equal Parts, as in the former 5 and from the Points Y p, &C. draw the right Lines I'M, p m, &e, parallel to C B ; and from the Points K, n % £&. the right LinesN M, n m, i$c. parallel to AC — The Points A M ;«, ESfo being connected, the Quadratrix isform'd ; wherein A B : A N : : AC : A P.
Here again, fince A B :AN :: AC: AP; if AB=s«, and AC = £, AN = a?, andAF=yj ax = by. See Qua- drature.
QUADRATUMCwfc", Quadrato-Quadrato-Qua- dratum, and Quadratum Stirdefolidi, ckc. arelNames ufed by the Arabs for the tftli, 81b, and 9th Powers of Numbers. See Powers.
QUADRATURE, Quadratura, in Geometry, the fquaiinsr, or reduction of a Figure to a Square 5 or the finding a Square equal to a Figure propofed. See Figure and Square.
Thus, the finding of a Square containing jufi as much Surface, or Area as a Circle, an Ellipfis, a Triangle, or other Figure, is call'd the Quadrature of a Circle, an El- lipfis, a Triangle, orthelike. See Circle, &c.
The Quadrature of Rectilinear Figures comes under the common Geometry j as amounting to no more than the finding their Areas, or Superficies 5 which are in effect their Squares. See Area.
Squares of equal Areas are here eafily had, by only ex- tracting the Roots of the Areas thus found 5 and on fuch Root as a fide conftructing a Square. See Square. See alfo the particular Methods of finding the Areas or Squares, under each particular Figure, asTai angle, Parallelo- gram, Trapezium, &c.
The Quadrature of Curves, that is, the meafuring of their Area, or the finding of a rectilinear Space equal to a cur- vilinear Space, is a Matter of much deeper Speculation 5 and makes a part of the higher Geometry. See Geo- metry.
Tho'thefiWiJ/^rfmre.efpeciallyofthe Circle, bee a thing many of the firft-rate Mathematicians among the Anticnts were Very fotticitous about, (fee Quadrature cf the Circle) yet nothing in this kind has been done fo considerable, as in and iince the middle of the laft. Century j when, viz. in the Year 1*57, Mr. Neil and my Lord JBrowiker, and afterwards, in the fame Year, Sir Chrijlopher Wren, Geo- metrically demonstrated the Equality of fome Curves to a Straight Line.
Soon after this, others at home and abroad, did the like in other Curves j and not long afterwards the thing was brought under an analytical Calculus, the firft: Specimen whereof ever publifhed was given by Mercator'm irjS8. in a Dcmonftration of my Lord Brounker's Quadrature of the Hyperbola by Dr. Wdllzs*& reduction of a Fraction into an infinite Series by Divifion. See Quadrature of the 'Parabola.
Tho' it appears by the way that Sir Jfaac Newton had heforedifcover'd a Method of attaining the Quantity of all quadrable Curves analytically by his Method of Fluxions, before the Yearrtfo'S- SccFluxions.
"Fis contefted between Sir Chrijlopher Wren and Mr. Huygem which of the two firlt Sound the Quadrature of any
determinate Cycloidal Space Mr. Leibnitz afterwards
found that of another Space ; and M. Bernoulli in itfo?, difcover'd the Quadrature of an infinity of Cycloidal Spa- ces, not only Segments, but alfo Sectors, £$c. See Qua- drature of the Cycloid, Qu a d r a t u r e of t he Zme, ckc Quadrature of the Circle, is a Problem that has em- ploy'd the Mathematicians of all Ages ; but Still in vain. See Cieclf..
It depends on the Ratio of the Diameter to the Periphe- ry, which was never yet determined in precife Numbers. See Diameter, ($?<?,
( 92-1 )
Q.UA
Were this Ratio known, (which would I'mnly the Cir- cumference's being exprefs'd by fome Affection of the Diameter ; and, of confequsnee, that it were equal to a right Lme) the Quadrature of 'tis Circle were effected.- it being demonstrated, that the Area of a Circle is equal to a rectangular Triangle, whofe two Sides comprehending the right Angle, are the Radius, and a right Line equal Co the
Circumference So that to fquare the Circle, all that is
requ.red ,s to reaify it. See CiRctjMFERENCEand Rec- tification.
Many have approach'd very near this Ration— -yft-o&i-
feems to have been one of the firft who attempted it ; which he did by means of regular Polygons inferibed and circumkribcd; and by ufing Polygons of jiff fides fixed the Ratio as 7 to 22. See Polygon.
Some of the Moderns have come nearer, particularly Lud. aCenlen, who with infinite Induftry found, at length, that fuppofing the Diameter r, the Circumference is lefs than 3-I4I 59*«535§5>79323S4<;2<>4333jS79 50; but yet grea- ter than the fame Number, if the laft Cypher be turn'd into an Unit.
Strifl Geometry here failing, Authors have had recourfe to other Means ; and particularly, to a fort of Curves, call'd Qtiadratrices : But thefe being Mechanical Curves, inltead of Geometrical ones, or rather Tranfcendental inftead of Al- gebraical ones, the Problem is not fairly folved thereby. See Transcendental.Mechanical, e?c. Quadratrix,
Hence, recourfe has been had, by others, to Analytics — and the Problem attempted by three forts of Algebraic Calculations — The ifi gives a kind of tranfcendental Qua- dratures, by Equations of indefinite Degrees: as if X* + a; be equal to 30, and x be fought, it will be found to be 3 ;
becaufe 3'+ ;, is 17 + 3, or 30. The id by vulgar
Numbers, tho' irrationally fuch ; or by the Roots of com- mon Equations, which for the general Quadrature, or its Seaors, is impoflible The 3d by means of certain Se- ries, exhibiting the Quantity of a Circle by a Progreflion of Terms.
_ Arithmetic, in effect, affords us very accurate and intel- ligible Expreffions for all rational Numbers; but it is de- feaive as to Irrationals, which are infinitely mote nume- rous than the former : there being, e.gr. an Infinity of 'em between 1 and 2. The Root of 2, which is a mean Propor- tional between 1 and 2, is a very obfeure Idea; and its Magnitude is fuch, as that if you would exprefs it in ratio- nal Numbers, which alone are clearly intelligible, you may ftill approach nearer and nearer its exaa Value, but never arrive precifely at it.
Thus, if for the Value of the Root of 2, you firft put 1, 'tis vifibly too little ; if, then, you add i, 'tis too much ; for the Square of 1 -J-i, or ofi, exceeds 2. If, again, you take away f, you'll find you have taken too much ; and if you'll return jj, the Sum willbe too great — Thus, may yon proceed to Infinity,without ever finding a Numbertoftop at.
Now thefe Numbers, thus found, being difpofed in their proper Order, make what we call an Infinite Series. See Series.
Further, of infinite Series's, there are fome which only yield a finite Sum, as 4, $, f , £gc. and in general all fuch as decreafe in Geometrical Progreflion — And there are others on the contrary, which make an infinite Sum ; as the Har- monical Progreflion, £, f ^, gfc. See Progression.,'
But, here, we have only to do with the former, as' ex- preffing a finite Magnitude ; yet cannot even the Sum of
thefe be always found Thus, we are certain, that 'tis
impoflible to find the Sum of the ,'ieries exprefling the Root of 2.
Geometry, however, is free from the Impoflibility A- rithmetic labours under, of exprefling irrational Numbers.
Thus, the Diagonal of a Square, whofe Side is 1,
expreffes the Root of 2. See Diagonal.
let in other Magnitudes, Geometry, itfelf, may fall un- der the fame Difficulty with Arithmetic For it is
poffible, there may be right Lines which cannot be ex- prefs'd but by an infinite Series of fimilar Lines, whofe Sum it may be impoflible to find.
In effea, the right Lines, which (Tiould be equal to
Curves, are frequently of this kind In fearching, e.gr.
for a right Line equal to the Circumference of a Circle, we find that the Diameter being put 1, the Circumference will be 4 lefs, I more, | lefs, f more, f, (Sic. making an infinite Series of Fraaions, whofe Numerator is always 4, and the Denominators in the natural Series of the uneven Numbers; and all thefe Terms, alternately, too great and too little.
Could the Sum of this Series be found, it would give the Quadrature of the Circle ; but this is not yet done ; nor is
it at all probable it ever will be done That, however,
is not yet demonftrated ; nor, of confequence, is the Qua- drature of the Circle yet demr-nftratpd impoflible.
To this it may be added, that as the fame Magnitude may be exprefs'd byfeveral different Series, 'tis poffible the Circum- ference of the Circle may be exprefs'd in fome other Series,
whofe
