QUADRATURES, METHOD OP. 



QTJ MTOR. 



the third, fourth, and fifth, and to on, the number of equal parts being 

 nv*. The approximate area U then found by adding to the first and 

 kit oflhet twice the sum of all the other odd one* and four tiuiM the 

 sum of all the even one*, and multiplying the sum *o obtained by the 

 thini part of the common base, or smallest subdivision of the whole 

 baie. The following method give* any required amount of approxi- 

 mation. Any integral [I.rmniATio*] con be found approximately by 

 a summit ion, the limit of which is the exact value : thtu we could 

 determine frdx from f a to jc a + h by dividing A into a large 

 number, H, of equal parU, and actually summing 



A / A\ A / A\ A / A\ A 



I'ut the special object of that which is called the method of quadrature* 

 U the perfection of this method of quadrature* (which is either very 

 inexact or very tedious) by subsequent corrections, which ore most 

 readily derived from such mathematical considerations aa ore described 

 in the article OPERATION. The theorem ou which it is founded may 

 be seen in p. 313 of the 'Differential Calculus,' in the ' Library of 

 Useful Knowledge : ' exhibited in the beat form for practical use, it in 

 a* follow* : 



Letyjr<&r be required to be found from x = ato x=a+k: divide A 

 into it equal parts, and let .\ a , A,, A, .... A, be the values of y cor- 



responding to the following value* of x, namely, o, a + -, a + 2 -, . . . 

 a-m- (or a + A). Compute 



Ami, writing down A,,, A,, Ac., as below, take their difference*. 



AO A, A, A 3 ....... A._j A._. A._l A. 



AA,, AA, AA, ......... AA._J AA..J AA_I 



A'A,, A'A, ........... A'A_, A'A_ 2 



AX ......... A'A.-J 



up to the fifth differences, which will be generally sufficient. Then 

 the value ulJyiLc from .t=ato.r = <n-/ia8 follows, very nearly : 



-; (A.+A )- T 2 - (AA.-.-AA.)- 

 .Ji ~ (A'A_,+ A'A,,)- ^ - ( AA_ - A'A,) - 



3 A 863 A 



X)- 



/dx 

 fromz=9 to Jt=13 



(we take purposely a case in which verification is easy.) Hera o = 9, 

 i 4, y-1 : x. Divide the interval 4 into ten equal parts, so that 



A : = -4. We have then 9,9-4,9'S, 126, 13, for the eleven values 



of x, and writing down their reciprocals, and taking their differences, 

 we have 



s 1-01354327 x -4 - -40541731 

 A + A. = -18803419 

 AA, - AA,, - -00228613 

 A'A. 4- A'A, - '00054610 

 A'A, - A'A,, - -000021*12 

 A'A. T A'A,, = -00000918 



Apjuvximate value of J" m -J*7W479 



The same result, found by common methods, is hyp. log. 18 hyp. 

 log. 9, or 2-M4949M - S'19722458, or -80772478 : so that the pre- 

 ceding method U in this case more than correct to seven figure* by 

 use of four difference*. 



This method of quadrature* is the mathematician's last resource 

 when all others fail or are still longer. In most of the case* in -hi.-h 

 it is absolutely requisite, the calculation of the value* of y 

 most tedious part of the operation. \Vlu-n the limit* are very ilifli-- 

 rent, it is generally requisite to divide their interval into several parts, 

 and to make the integrations through the several part* separately. 

 For an inverse prooea* see SUMMATION. 



QUADRICHLORACETAMIDK, or OMoracrlamie arid (C lie! 



. f" \ 



NO, = N^ Cl L is a product of the action of chlorine mum 



I <'.<', O,/ 



trichloracetauiide. It occurs in colourless, inodorous, aciculnr prisms, 

 of disagreeable taste. 



QUAJDRICHLOROBUTTBIC ACID. [BUTYRIC ACID.] 



QUJJSTOR (from nmm: " r/tii eonqninrrt /niMinit prcntilia et 

 malejlcia," Varro, ' De Ling. Lat.,' Iv. 14) is a name which was coriinn'ii 

 to two distinct classes of officers at Rome, who were only distinguished 

 from each other by different attribute* : the name of one class w*s 

 qucettora parrieidii ; that of the other, qutrttoret clanrici. 



It is said by some that the office of i/utrttara //arn'nVii existed even 

 in the reigns of Romulus and Numa, Livy (i. 20) and T 

 (' AnnaL,' xi. 22) think that they were appointed by the kings; but it 

 is more probable that the kings only proposed the candidates, and that 

 they were appointed by the populus. (Ulpian, ' De offlc. (,> 

 1 Dig.,' i. 13.) That the office existed in the reign of Ttillu ! I 

 is certain, and the general opinion among the Romans was tl:.it 

 instituted by that king. After the establishment of the n-piiMu-, tin- 

 two qusstore* parrieidii continued to be elected in the com 

 the curies, on the presentation of the consuls, as they were before on 

 that of the kings : but they were now regularly elected every year, 

 whereas before they had only been appointed in cases of cine! - 

 After the decemvirate, they were elected by the centuries. At tint 

 time of the Licinian law one part of their functions was swallov 

 by the office of the triumviri capitalus, while the rest were embodied 

 in the offices of the curule cediles and the tribunes. 



The ijitattores parrieidii, according to Nii-Uihr, were the name as the 

 duumviri perdutllionu, but Walter (' Gesch. des Rom. Rechts,' p. 855) 

 has adduced a number of passages which seem clearly to prove that 

 the ijiKrxtarat parriciilii must be distinguished from the <l 

 dutttiottii, who continued to be elected to the end of the republic, and 

 were real judges in cases of perduellio. The quacstores, on the other 

 hand, were a kind of public accusers, who conducted the accn 

 and carried the sentence into execution. (Festus, s. v. ' Parici ;' 

 Dionys. HaL, viii. 78.) 



The qiutitorcs <-/<unVi'had the superintendence of the public treasury, 

 and are said to have been instituted by Valerius Publicola. who gave 

 the right of electing them to the populus, though Wachsmuth 

 (' Geschichte des Ronusches Stoat*') contends that there was only . un- 

 kind of quicstor. At first they were only two in number, and in the 

 year 421 B.C. their number was doubled (Liv., iv. 43.) and part of 

 were to be plebeians, Imt this was not the case until ten yearn 

 wards, when three out of the four qiiicstorea were plebeians. (I. ivy, 

 iv. 48, S4.) From the time that four qmestores were elected, two 

 accompanied the consuls into the field, while the two others remained 

 in the city (quasstores urbani). After the Romans had made them- 

 selves masters of all Italy (4S9 n c.), the number of quaestore* was 

 again doubled, so that there were now eight of them (F.iv., ' Kpit..' lil>. 

 xv.), for the adminixtr.-ition of the financial affairs in the city, in th,- 

 army, in Italy, and the province of Sicily. One of them, who i 

 at Ostia, had ;Jso to provide the city with corn. (Cic., ' pro Sext.,' 

 17.) Sulla in his dictatorship raised their mimU-r to twci.i- 

 Cscsar to forty. (Tacit., 'Annal.,' 1. c. ; Dion Cass., xliii. I. 

 During the time of the emperors their number v.uii'.l. The two 

 quicstores urbani, down to the time of .Inliim Cicsar, had the adminis- 

 tration of the public treasury ; they registered the revenue and 

 expenditure of the republic (Ascon. Fed. on Cic., ' in Vert.,' ii. 1. p. 

 158, ed. Orclli; Pint., ' Cat. Min.,' 17, 18). receive.! the money due to 

 the state, and mode the payments sanctioned by the senate. They 

 had also to receive and take under their especial protection all 

 foreign ambassadors, and those strangers who were connected with 

 the state by ties of public hospitality ; finally they had the care of 

 the funerals and monuments which the senate decreed as distinc- 

 tions for men of great merit (Plut, ' Qucost Rom.,' 48 ; Val. Max., v. 

 1,1.; Cic., ' pro Flacoo,' 18) : they kept in the treasury the books in 

 which the senatus cousulta were copied, until Augustus also entrusted 

 them with the keeping of the original .Wuim-nts. (Dion Cass., liv. 

 86.) Julius Caesar transferred the administration of the treasury 

 the .piii-t. TO* to two rodiles. (Dion Cass., xliii. 47.) 



The military qutcstores who accompanied the consuls into the fi.-l.l 

 (Cic., 'in Verr.,' ii. 1. 15) had the charge of the money with hie|i 

 the war was carried on, distributed among the soldiers their j.r. 

 anil pay, and superintended the aale of the booty, the product- of which 

 was cither divided among the soldiers or lodged in tin- puMic: treasury. 

 (Livy, iv. 53.) They had however to give in an account of all 



