TAX IX 



TAYI.OIfS THEOREM. 



Reformation. and enabled them very commonly to obtain larger lubsi- 

 <ii<M from the convocation than those that were voted by parliament. 

 The church, therefore, wu not unwilling to be deprived of the expen- 

 sive privilege of voting separate auDaidies; and acquiesced in an 

 arrangeoMat propueed in lb4-5, by which the Common* have ever 

 since voted Uxe* upon the possessions of the church and of the clergy, 

 in the same manner a* upon the laity. 



TAXES. The general objects, character, and principle* of taxation 

 and of different classes of tax**, an treated of under the head of TAX, 

 TAXATION. In this place it i* propoaed to give a abort summary of the 

 amount and description of Uxe* paid in Great Britain and Ireland, 

 whether assessed directly upon property, or collected indirectly upon 

 article* of consumption ; including not only such taxes a* are paid to 

 the general government, but alao all municipal and local asaeasmenU or 

 contribution*. 



I' HI ted 



The chief (ourae* of revenue will be seen by the following statement, 

 made up to December 31, 1860. 



Groi Receipts. 

 Ourtoms ....... 13,031,399 



ExelM ....... 19,069,000 



Sumps ....... 8,2i,198 



Taxes (land and asMsacd) . ... S, 126,000 



Property tax ...... 19,801,816 



Port Office ....... 3,420,000 



Crown land* ..... 289,968 



Miscellaneous ...... 1,843,457 



Total revenue 



.71,967,491 



From which the salaries, &c. of the revenue departments, amount ing 

 to 4,509,402/. ha* to be deducted as the charge for collecting; of this, 

 the Poet Office expenses for management in 1859 was 1,851,953/. The 

 remainder, 2,657 ,449/., gives an average of lees than 9rf. in the pound as 

 the charge of collection. 



The *SPBsed taxes are the house-tax, the tax on male servants, 

 taxes on carriages, on horses, on dogs, armorial bearings, horse-dealers' 

 duty, game duties, stage-coach duties, and duties on passengers con- 

 veyed for hire by carriage* travelling on railways. In 1840 (3 & 4 

 Viet. c. 17) 10 per cent, additional was imposed on all the assessed 

 taxes. The income-tax was imposed April 5th, 1842, for three years, 

 and was then renewed for another three years, and has been continued 

 at varied intervals ever since. For 1880 it was imposed at lOrf. in the 

 pound ; for 1861-2 it is proposed to reduce it to 9d. in the pound. 



Bachelors, except Roman Catholic clergymen, pay an additional 

 duty of M. on male servants. [BACHELOR.] The charges for game 

 duties are stated under GAME LAWS. The duty on passengers con- 

 veyed for hire by carriages travelling on railways is 5 per cent, on 

 the gross amount of the fares. As to the duties on stage-coaches, see 

 STAGE-CARRIAGE. 



To these parliamentary taxes may be added the following local 

 assescmenta in England and Wales : 





 Poor-rates (which include* county, borough and ) . 



police rate., 700.000/.) . . . . ] 8 ' 188 . 88C 

 Church-rates (average of seven years) . . . 261,716 

 Additional voluntary contributions . . . . 269,980 



1 1 iirb way-rates 1,949,817 



Turnpike-lolls (England and Wales) . . . 1,031,090 

 Metropolitan local management .... 199,886 



The rates levied for local management could not be ascertained, but 

 including poors-rate* in Scotland and Ireland, in the grand jury pre- 

 sentment* in the latter country, the whole is estimated in a return, 

 avowedly imperfect, laid before parliament, at 15,171,6462. in 1860. 



The tithes of Great Britain and Ireland are said to amount to 

 4,000,000*. 



There was published under the direction of the Poor-Law Com- 

 missioners in 1846, a valuable work entitled ' The Local Taxes of the 

 United Kingdom, containing a Digest of the Law with a Summary of 

 statistical Information concerning the Several Local Taxes in England, 

 Scotland, and Ireland.' England includes England and Wales, It is 

 remarked in the Introduction that " these Local Taxes are of two kinds : 

 the rates raised in defined districts ; and the tolls, dues, and fees paid 

 for particular services or on certain occasions. But those rate* only 

 will be here noticed, which are authorised by general statutes or the 

 common law; excluding such as derive their origin from special 

 or local Acts." The rates are divided into three classes. I. Rate* 

 of independent district*, on the basis of the poor-rate. II. Rates of 

 independent districts, not on the basis of the poor-rate. III. Rates oi 

 aggregate district*, on the basis of the poor-rate. No. I. comprehends 

 1, The Poor Rate. 2, The Workhouse Building Rate. 3, The 

 Survey and Valuation Rate. 4, The Jail Fees' Rate. 6, The Con- 

 stables' Rate. 6, The Highway Rates (three). 7, The Lighting and 

 Watching Rate. 8, The Militia Kate. >'... II. . ,,i, lp i.'hcn(U 1. Tin- 

 Church Rates (three). 2, The Sewer Rate. 3, The General Sewers' 

 Tax. 4, The Drainage and Inclocurc Rate*. 5, The Inclosure Rate. 

 6, The Regulated Pasture Kate. No. III. comprehends Comtiri. 1, 

 The County Rate. 2, The Police Rate. 3, The Shire Hall Rate. 4, 



The Lunatic Asylum Rate. 5, The Burial Rate. B inutotb. 6. The 

 Him.ir.il Rate. Bonmgla. 7, The Borough Rate. 8, The Watch 

 Kate. 9. The Jail Rate. 10, The Prisoners' Kates. 11, The Lunatic 

 Asylum Rate. 12, The Museum Kate.- -Coiattitt and Borwight. 13, 

 The District Prison Rates. 



The Local Taxes in Scotland are distributed by Mr. Burton under 

 the following heads : 



I. Administration of Justice, which includes Criminal Prosecutions, 

 Court Rooms and County Buildings, Rural Police, Town Police. Prisons. 

 II. Internal Transit, which includes Commutation Roadi, T 

 Roads, Highland Roads and Bridges. III. Navigation. \\ 

 Economy, which includes Direct Municipal Taxes -toiu, 

 Miscellaneous Burdens. V. Relief of the Poor. VI. The Church and 

 Education, which includes The Church of Scotland Education. VII. 

 Miscellaneous Taxes. 



Mr. Burton observes " that the money expended on the ecclesiastical 

 establishment and on education, partakes, in some respect*, of the 

 nature of a tax." The amount of money annually levied by local 

 taxation in Scotland is not accurately known. The sum of 956,6782. is 

 the approximate amount given by Mr. Burton. 



The Local Rates levied in Ireland are distributed un.l- T 

 heads in the work published under the direction of the Poor-Law 

 t'ommisaioncrs. 



I. Grand Jury Cess (in all the counties, including counties of 

 cities and towns). 



II. Poor-Kates (in 130 Unions, comprising every townland and 



denomination of land in Ireland). 



III. Lighting, Cleansing, and Watching Rates (in all cities, towns, 



and boroughs which may adopt the provisions of the 

 statute). 



IV. Borough Rates (in certain Boroughs). 



V. Pipe Water Kate* (in every city and town, except Dublin, 

 Cork, and Limerick, which gives title to a bishop or arch- 

 bishop). 

 VI. Parish Cess (in all parishes, unions of j parUhro, or chapel rics in 



Ireland). 

 VII. Rates for deserted children (in all parishes in Ireland, except 



those in the city of Cork). 



V 1 1 1. Ministers' Money (in cities anil towns corporate in Ireland i. 

 IX. Board of Health Rates (in parishes in which the lord In' 



shall direct officers of health to be appointed). 

 "Besides the above rates leviable under general acts of pail. 

 there are rate* leviable under special acts in many places, a* Dublin, 

 Cork," Ac. 



Information about the several taxes of European States will be f "in. 1 

 in the Parliamentary Paper, No. 227, of 1842, ordered by the House of 

 Commons to be printed, 3rd May, IS i J. 



TAYLOR'S THEOREM. We propose in this part of the art 

 give gome results of the methods of algebraical development which are 

 consequences of the celebrated theorem, the history of which is given 

 in the article TAYLOR, BROOK, in Bioo. Div. The simplest parts of the 

 Differential and Integral Calculus will be presuiiicil known. It in not 

 usual in works on that subject to bring together in one place the 

 most conspicuous theorems which have arisen out of that of Taylor ; 

 which makes it the more desirable that such a thing should be done 

 in a work of reference. It is to be particularly remembered that we 

 do not here profess to teach the subject of development, but only to 

 recall the steps of the several processes to those who have already 

 learnt them, and to present the theorems in a form which can be easily 

 referred to. 



As to notation, we shall frequently signify differentiation by accents: 

 thus <f"x is the second diH'rivntial coefficient of *.r with respect to z; 

 (0.nf>.r)"' is the third differential coetlicient of the product of <p.c and 

 ^.r. And [] will signify the product Ix2x3x . . . . x ( 1 ) x H. 

 Moreover, when a series is written, three terms will be written down, 

 and the general term appended. 

 Taylor's theorem is as follows : 



/<' r A* 



<j,(x + h) = <fjf + (f>'x . h + $"* T + ic. 



This theorem is true whenever x has such a value that 1. No one 

 of the set ^.r, <)>'.r, ic. is infinite. 2. All of them do not vanish. Thus 

 neither of the following could be allowed to be treated by it \\ln-n 

 jc=o : 



V(^ - o) . log * and -<"' '. 



In the first function, f'x, and all which follow, are infinite when 

 x=a; in the second ^r and all its differential coefficient* van inn when 

 x = o. The meaning of this circumstance is as follows : the form of 

 Taylor's theorem essentially requires that <f>(j- + A) should bo developed 

 in ascending integer powers of A ; consequently when such form of 

 development a impossible, this theorem may show signs of being 

 inapplicable. Now, the first of them functions (when x= a) can only 

 have f (a -f A) expanded in ascending fractional powers ; and the 

 only in descending integer powers. Those who will only allow tin- 

 use of converging series may require also that A should be so small that 

 the resulting series is convergent : but this Cauchy has proved always 

 happens if A be small enough. 





