QUA 



cumference of circles, or other curves, and of the fcvcral 

 parts of it. 



Or, more accurately, the quadratrix of a curve is a 

 tranfcendental curve defcrihed on the fame axis, the femi- 

 ordiiiates of which being jriveii, the quadrature of the cor- 

 refpondent parts in the other curve is likewife given. See 

 CUKVE. 



Thus, e.gr. the curve AND {Plate XIII. Jnalyjis, 

 Jig. 7.) may be called the quadratrix of the parabola A M C, 

 fincc it is demonllrated, that A P M A is = P N , or 

 APMA = APx PN, orAPMA=PNxa, a 

 cor.ft.ant quantity, &c. 



The mofl. eminent of thefe quadratrices are, that of Di- 

 noftrates, and that of Mr. Tlchirnhaufen, for the cirqle ; 

 that of Mr. Perks for the hyperbola. 



QuADKATlllX of Dmojlrales, is a curve A M m m [fig. 8.) 

 by which the quadrature of the circle is effcfted, though not 

 geometrically, but mechanically ; it is thus called from its 

 inventor Dinoltrates. 



Its genefis is thus : divide the quadrantal arc A N B into 

 any number of equil parts, in N, n, &c. by a continual bi- 

 feAion ; divide the radius A C into the fame number of 

 parts in the points P,/>, &c. Draw radii C N, C «, &c. 

 Laftly, on the points P, /I, &c. credl perpendiculars P M, 

 i m, &c. The curve formed by connedting thefe lines is the 

 quadratrix of Dinoilrates. 



This curve may be defcribed by continual motion ; if we 

 fuppofe the radius C N by its extreme N to defcribe uni- 

 formly the arc A B, and at the fame time a ruler P M, always 

 parallel to itfelf, to move uniformly along A C, in fuch a 

 manner that when the ruler P M arrives at C, the radius C N 

 may coincide with C B ; and thus the continual interfeftion 

 of C N with the ruler P M will defcribe the quadratrix 

 AMD. 



Here, from the conftruaion, ANB:AN::AC:AP; 

 and therefore, if A NB = a, A C = *, A N = .v, AP =_)r ; 

 ay = i X. See Quadrature. 



Quadratrix Tfchirnhauftaim, is a tranfcendental curve 

 A M m ;« B (Jig. 9.) by which the quadrature of the circle 

 is likewife etfefted ; invented by Mr. Tfchirnhaufen, in imi- 

 tation of that of Dinoftrates. 



Its genefis is thus conceived ; divide the quadrant A N B, 

 and its radius A C, into equal parts, as in the former ; and 

 from the points P, p, &c. draw the right lines P M, p m, 

 &c. parallel to C B ; and from the points N, n, &c. the 

 right lines N M, ;; m, &c. parallel to A C. The points A, 

 M m, being connefted, the quadratrix is formed ; in which 

 ANB:AN::AC:AP. And therefore, if A B = .z, 

 and A C = i^, A N = x, and AP — y; ay — iz. See- 

 Quadrature. 



This curve may be alfo defcribed by continued motion, if 

 two rulers, N M and P M, perpendicular to each other, be 

 made to move uniformly and parallel to themfelves, the one 

 along the quadrant of the circle A C, and the other along 

 the radius. 



QUADRATUM-CUBI, Quadrato-quadrato-qua- 

 dratum, and QuadratUiM SurilnfoUdi, &c. are names ufed 

 by the Arabs for the fixth, eighth, and tenth powers of 

 numbers. See Power. 



Quadratum Os, in Comparative Anatomy, os carre oi the 

 French ; a fmall bone in the head of birds, to which the 

 lower mandible is articulated. See Birds, in Comparati-ve 

 Anatomy, in the divifion relating to the bones. 



QUADRATURE, in Geometry, fignifies literally the 

 finding of a fquare equal in area to any given figure, w'hich 

 was the method the ancients made ufe of when they had 

 in view the determination of the furface of any fpace ; but 



QUA 



the term quadrature has now a more indefinite fignificntion ; 

 implying, in general, the determination of the area of a figure, 

 without any reference to the geometrical exhibition of it, in 

 u fquare or other rectilinear form. 



AH rcftilinear figures being immediately reduced to, or de- 

 pendent upon, the area of triangles, tlieir quadratures 

 have been known from the highell antiquity ; but the quad- 

 ratures of curvilineal fpaces are, with very few exceptions, of 

 modern date, two only having been known till near the be- 

 ginning of the eiglitcenth century. 



Tile fir (I curvilinear fpace whofe quadrature was accu- 

 rately determined, waa the lune of Hippocrates, of which an 

 account will be fousd under the article LuNK. Archimedes 

 next found the area of the common parabola ; which he ob- 

 tained in a very ingenious manner, by inlcribing an ifofceles 

 triangle in the parabola, then two ifofceles triangles on the 

 equal fides of the former, four others on thefe, and fo on, 

 which he found to have a certain relation, decreafing in the 

 proportion I, 5, tV> &c. the infinite fum of which feries 

 would therefore exprefs the area of the parabola, or the 

 area of all the triangles of which he thus conceived it to be 

 compofed ; and which fum he found to be i J^ or 4 of the 

 circumferibing redtangle. After this time, a period of near 

 two thoufaud years elapfed, without producing the quadra- 

 ture of a fingle curvilinear figure, although the fubjeft feems to 

 have engaged the attention of the moll eminent mathemati- 

 cians during that long interval, particularly the quadrature of 

 the circle. This figure, being the moil fimple in appearance 

 and conSruiJtion of any contained under a curve line, was 

 well calculated to excite the curiofity of mathematicians. 

 Archimedes doubtlefs attempted the folution of this problem ; 

 but failing in producing the exaft quadrature, he contented 

 himfelf with giving an approximation, (hewing by the infcrip- 

 tion and circumfcription of a polygon of ninety-fix fides, 

 that the diameter being i, the circumference was greater 

 than 3J4, but lefs than 3f; ; and as it was knov^n, even 

 before the time of Archimedes, that the area of a circle is 

 equal to that of a right-angled triangle, whofe altitude is 

 equal to the radius, and bafe equal to the circumference of 

 the circle, it follows, that the area would be greater than 

 I j-j, but lei's than : ^4. 



It would be ufelefs to attempt in this place to enumerate 

 the various abiurd quadratures which have been, from time to 

 time, publilhed by minor geometers, with all that conceit 

 and confidence which feldom fail to accompany inferiority. 

 Some attributed their fuccefs to divine inlpiration ; others to 

 their own fuperior talents : fome offered large fums of money 

 to thofe who (hould difcover any error in their inveltigation, 

 while others expelled great rewards from their government, 

 as a recompence for their difcovery, foolilhly attaching great 

 importance to a problem, which, if it could be accurately 

 folved, would ferve no other purpofe but to gratify the cu- 

 riofity of mathematicians. Many of thefe attempts, how- 

 ever, have been rendered fomevvhat amTifing by an excefs of 

 abfurdity. This is particularly the cafe with regard to the 

 work of Jaime Falcon, a Spaniard of the order of Notre 

 Dame, of Montefa, publifhed at Anvers in 1587. This 

 treatife opens with a dialogue in vcrfe between himfelf and 

 the circle, which thanks him very affetliunately for having 

 fquared him ; but the good and model! knight attributes all 

 the honour of the difcovery to the holy patron of his order. 

 SeeMontucla's " Hilloire des Recherches fur la Quidrature 

 du Circle ;" or his " Hiftoire des Mathematiques," vol. iv. 

 p. 619. 



Referring tliofe readers who have the curiofity to examine 

 the reveries above-mentioned to the two preceding works, 

 we propofe to give here an abftradl from the fame, of what 



has 



