R E C 



R E C 



R>0 



nt"- 



" (" - ,»-, 



+ &C. 



^ nt"- ' — —^ '-(«-- — &c. 



+ arc 



So that, generally, arc to tan. i = » x arc to tan. — 



If we take ti = i, and ; = 3, we have arc 



2, and / = 3i 



to tan. 



T" 



tan. I = arc tan 



— 4- arc tan. — If « 

 3 ^ 



we have arc tan. i = 2 x arc tan 

 « :::= 4, and / 



I I 



— 4- arc tan. — 



3 7 



If 



5, we have arc. tan. i = 4 x arc tan. — — 



arc tan ; which is tlie formula of Machin ; and by 



239 

 giving other values to « and /, a variety of other formulae 

 might be found, though it would, probably, be difficult to 

 find one more convergent than the laft. The reader will find 

 more on this fubjeft in vol. i. of Dr. Hutton's Trafts. 



We fliall conclude this article by giving the circumference 

 of the circle to 155 places, as given by Zach, from a ma- 

 nufcript which he faw in the Ratcliff library at Oxford, dif- 

 tinguifliing the periods of the feveral npproximatiens of dif- 

 ferent authors mentioned in the preceding part of this article, 

 and the article QuAnuAXUKE. 



The diameter pf a circle being i, the circumference will be 

 a b c d 



3.14 159, 2 6;35, 89793, 2 3846, 



e 

 26433- ^3279' 50288, 4'97i. 



69399> 37510' 58209, 74944> 



f 

 59230, 78164, 06286, 20899, 



g 



86280, 34825, 34211. 70679, 



* 

 82148, 08651, 32823, 06647, 



h 

 09384, 46095, 50582, 23172, 



i 

 53594, 08128, 4802. 



a, Archimedes ; b. Melius ; c, Vieta ; d, Adrianus 

 Romanus ; e. Van Ceulen ; f, Abraham Sharp ; g, Ma- 

 chin ; h, Lagny ; i, Oxford Manufcript. 



* This figure is a 7 in Lagny's approximation, but Vega, 

 in the revifion of the computation, afl'erts that it ought to 

 be an 8, as we have given it. 



RECTIFIED Spirits, &c. are fuch as have undergone 

 the operation of reftification, or have been diftilled over and 

 over, to feparate from them any heterogeneous matter, 

 which might have arifen with them in the former diftilla- 

 tions. 



Hence we fay, fpirit of wine twice reftified, thrice rec- 

 tified, &c. ^ 



It is the reftification that makes the difference between 

 brandy and reftified fpirits of wine. See Spirits. 



RECTIFIER, in Navigatien, is an inftrument ufed for 

 determining the variation of the compafs, in order to rectify 

 the {hip's courfe, &c. 



It confifts of two circles, either laid upon, or let into 

 one another, and fo faftened together in their centres, that 

 they reprefent two compades, the one fixed, the other 

 moveable ; each is divided into thirty-two points of the 

 compafs, and three hundred and fixty degrees, and num- 

 bered both ways, from the north and the fouth, ending at 

 the call and well in ninety degrees. 



Tlie fixed compafs reprefents the horizon, in which the 

 north, and all tiie other points, are liable to variation. 



In the centre of the moveable compafs is fallened a filk 

 thread, long enough to reach the outfideof the fixed com- 

 pafs : but if the inllrument be made of wood, an index is 

 ufed inilead of the thread. 



Rectifier, in the DyilUcry, the pcrfon whofe employ- 

 ment is to take the coarfe malt-lpirit of the malt-lliller, and 

 re-di(lil it to a finer and better liquor. The art of the rec- 

 tifier might be entirely fet afide, if the malt-fliller could 

 make his fpirit perteft at a fecond operation ; which feems 

 very prafticable, if tlie malt-llillers could be induced to 

 forfake their old practice. The great tilings to be recom- 

 mended for the improvement of their art, would be, firft, 

 the brewing in perfedlion ; and fecondly, the keeping of 

 their wafii after the manner of Hale beer, till it has entirely 

 loll its malt flavour, and acquired a pungent acid vinofity ; 

 and then, thirdly, leaving out the lees, to diflil with a well- 

 regulated fire. It is fcarcely to be thought how pure a 

 fpirit is to be obtained from malt this way ; but the great art 

 would be, the finding of a method to make malt liquors arti- 

 ficially flale, bright, and flavourlefs, though otherwife vinous. 

 Shaw's Leftures, p. 223. 



RECTIFYING of Curves. See Rectification. 



RecTIPYIxg of the Globe or Sphere, is a previous adjuft- 

 ing of the globe or fphere, to prepare it for the folution of 

 problems. For the method of doing it, fee Ufe of the Ce- 

 kfl'ial Globe. 



RECTILINEAR, Right-lined, in Geometry, is ap. 

 plied to figures whofe perimeter confills of right lines. 



Rectilinear Angle, Maps, and Superjicks. See the fub- 

 flantives. ' 



RECTITUDE, Rectitudo, Reaum, in matters of 

 philofophy, refers either to the aft of judging, or of will- 

 ing ; and therefore, whatever comes under the denomination 

 of reftitude is either what is true, or what is good : thefe 

 being the objefts about which the mind exerciles its two 

 faculties of judging and willing. 



Recl'ttude of the mind, confidered as it judges, i. e. recti- 

 tude of the faculty of judgment, confills in its agreement 

 and conformity to the nature and reafon of things, and in 

 its determming and deciding about them according to what 

 their conilitutions, properties, ufes, &c. really are. 



ReS'itude of the mind, confidered as it wills, called alfo mo- 

 ral reSitude, or uprightnefs, confills in the choofing and pur- 

 fuing of thofe things which the mind, upon due enquiry and 

 attention, clearly perceives to be good ; and avoiding thofe 

 that are evil. 



RECTITUDINES, in La-w, rights, or legal dues, be- 

 longing either to God, or man. See Right. 



RECTO, a writ ufuaOy called a writ of right ; of fo high 

 a nature, that whereas other writs in real aftion are only to 

 recover the pofleffion of the lands, &c. in queflion, lofl by 

 the plaintiff, or his ancellor ; this aims to recover both the 

 feifin thus loft, and the property of the thing : fo that both 

 rights are here blended together ; that oi property, and that 



If a man lofe his caufe upon this writ, he is without 

 all remedy. 



There are two kinds of this writ : breve magnum de reSo, 



or 



