R E C 



II ii C 



^ Tliere can be but one right angle in a plain triangle ; 

 therefore a rcftanglcd triangle cannot be equilateral. 



RECTANGULAR, in Geometry, is applied to figures 

 and folids which have one or more angles right. 



Such are fquares, reftangles, and reftangled triangles, 

 among plain ligures ; cubes, parallelepipedf, Sec. among 

 folids. 



Solids are alfo faid to be rrdnngular with refpcft to their 

 fituation : thus, if a cone, cyUnder, &c. be perpendicular 

 to the plane of the horizon, it is called a redtangular or 

 right cone, cylinder, &c. 



The ancient-B ufed the phrafe reHangular fetVion of a cone, 

 to denote a parabola ; that conic fedion, before Apollonius, 

 being only confulered in a cone, whofe feftion by the axis 

 would be a triangle, right-angled at the vertex. See Conic 

 SeSions. 



Hence it was that Archimedes entitled his book of the 

 quadrature of the parabola, by the name of " Rettanguli 

 Coni Seftio." 



Rectangulah Barometer. See Barometer. 



Rec'tangulak Whidmills. See Windmills. 



RECTIFICATION, compounded of redus, right, ill- 

 refl, and^fo, / became, the aft of reRify'mg, i. e. of correfting, 

 remedying, or redrelling, fome defctt or error, in refpeft 

 either of nature, art, or morality. 



Rectification, in ChemtJIry, is the repeating of a diilil- 

 lation or fublimation feveral times, and generally with a lefs 

 degree of heat than at firft, in order to render tlie lubftance 

 purer, finer, and freer from aqueous and earthy parts. 



Reftification is a reiterated depuration of a diililled mat- 

 ter, e. gr. brandy, fpirits, or oils, by diftilling them over 

 again, to render them more fubtile, and exalt their virtues. 



That the rcdtification of fpirits may, in all cafes, proceed 

 with the greatell exatlnefs, a due regard to it mull be had 

 even from the firft fermenting the fubllance from which they 

 are to be made, and continued through all the ftages of dif- 

 tillation, the low wines, proof fpirit, and alcohol. The 

 management of the fermented liquor, to this purpofe, is 

 principally the letting it ftand to fubfide after the fermenta- 

 tion is over, and the drawing it off clear and thin, not too 

 rich for the ftill. The ftill is not to be overfilled with this. 

 Great care mull be taken to prevent its burning, and the 

 faints that run laft muft be kept feparate. not mixed with 

 the refl of the liquor diililled, which is now called the low 

 wines. In the rettifying of thefe into proof fpirit, great 

 caution muft be ufed that the fire be kept regular, not raifed 

 by fudden fpirts, which always throw up the oil in large 

 quantities, which is to be left behind. In the fucceeding 

 reftification of the proof fpirit into alcohol, the fame cau- 

 tious management of the fire is necelTary ; and, in both this 

 and the laft, the faints are not to be fuffered to run in among 

 the fpirit, but to be faved feparate. They may be all mixed 

 together at laft, and reduced to a fpirit fit for burning in 

 lamps ; but the keeping out of the reftified liquor will keep 

 away the coarfeft and moil ftinking part of the oil of the in- 

 gredients. 



By thefe eafy means, without any additional trouble or 

 charge, we might be furniftied witli a fpirit greatly exceed- 

 ing what we commonly meet with. And in general, the 

 art and myilery of our fellers of the feveral forts of Englifli 

 brandies, fcem to confift in this prudent management, and 

 in the adding a little of the oleum -virii, or oil of wine-lees, to 

 the fpirits tliiis procured pure : this gives the flavour of 

 foreign brandies, and is fo exteiifive in its ufe, that half an 

 ounce of it is fufficient for a hogrtiead of pure fpirit. 



M.ilt fpirit is that which principally requires all this care 

 ill the redtitication, bccaufe its oil is more naufeous and 



oftenfive than that of any other fpirit ; but all others will bt" 

 greatly the better for being treated in the fame manner, 

 and it is indeed necefl'ary that they (hould for fome particular 

 afes. 



It is remarkable, that no one method of combinatory re<.- 

 tification, that is, of the rettification performed by mean;, 

 of fait, and other additions, is fuited to all the feveral kinds 

 of fpirits ; fcarcely indeed will any one way lerve for any t\A o 

 fpirits: but this method, by fimple and careful diftillation, 

 is equally fuited to all. Molades-fpirit, cydcr-fpirit, wine- 

 Ipirit, or brandy, rum, and arrack, are all improved by it ; 

 and all ol them are then known to be perfectly reftifitd, 

 when, in the ftate of alcohol, they not only prove totally 

 inflammable in a little veiiel floating upon cold waters, but 

 when poured into the purcll Ipring water they have not the 

 leaft power of making any change in it, nor leave any marks 

 of oihnefs, or that undluofity which, on the mixture of the 

 lefs pure fpirits, floats on the top, and in certain lights gives 

 the rainbow colours. Shaw's Eflay on Diftillery. See 

 Alcohol and Distillation. 



Fixed falts are re£lified by calcination, diflblution, or fil- 

 tration. 



Metals are reftified, i. e. reined, by the coppel ; and re- 

 guhifes by repeated fufions, &c. 



In a word, all rectifications are founded upon the fame 

 principle ; and confift in feparating fubftances more volatile 

 from fubftances lefs volatile ; and the general method of 

 effecting this is to apply only the degree of heat which is 

 necelTary to caufe this feparation. 



Rectification of intrwlk Add. See Concentration. 



Rectification, in Geometry, is the finding of a right 

 line equal to a propofed cnrve, or fimply finding the length 

 of a curve line ; a problem which, even in the prefent ad- 

 vanced ftate of analyfis, is attended in many cafes with con- 

 fidcrable difficulty ; and was, in all, totally beyond the 

 reach of the ancient geometers, who were not able to afTign 

 the length of any curve line whatever ; though they could, 

 in a few inftances, determine the area of a curvilinear fpace. 

 See Quadrature. 



The firft rectification of a curve line was effected by Mr. 

 H. Neal, as we are informed by Dr. Wallis, at the con- 

 cluCon of his Treatife on the Cilioid. This curve was the 

 femicubical parabola, and Neal's rectification of it was pub- 

 lifhed in July or Auguft, 1657 ; and two years after, viz. 

 in 1659, the fame was done by Van Haureat in Holland. 

 See Schooten's Commentary on Defcartes' Geometry. 



It is, however, to the dottrine of fluxions that we owe 

 the complete reftification of curve lines, in finite terms, 

 when they admit of it ; and in others by means of infinite 

 feries, circular arcs, logarithms, &c. ; of which method we 

 fhall give a general view in the prefent article. 



Let A M O ( Plate XIII. Analyfis, Jig. i o. ) be a curve of 

 any kind, whofe ordinates are parallel to one another, and 

 perpendicular to the axis A H ; and let the fluxion of the 

 abfcifs A P be denoted by P/> or M R, and R m be taken to 

 reprefent the correfponding fluxion of the ordinate P M, 

 then will the tangent M m be the line whicii the generating 

 point of the curve would defcribe, if its motion were to be- 

 come uniform at M ; confequently this line will truly ex- 

 prefs the fluxion of the fpace A M. Hence putting A P 

 = .V, P M = >■, and A M = z, we have z = M »; = 

 ^/ (MR' + Rm') =^ V (x' + -;-) ; from which, and the 

 equation of the curve, the value of z may be determined. 

 But if all the ordinates of the propofed curve ARM 

 (Jig. ) be referred to a centre C ; then, putting the tan- 

 gent R P, intercepted by the perpendicular C P, = /, the 

 arc B N, of a circle defcribed about the centre C, =; x, the 



radius 



