34 Mathematical Correfpondence. [ Julys 
and 12, we fhall have refpeively the numbers 25207, 62921, $0641, 108361, 136081, 
163801, 191521, 219241, 246961, 274681, 302401, 330122, which numbers will anfwer the 
firft-condition of the queftion. 
Again, let y reprefent the leaft number, which will anfwer the fecond condition: then, per 


; yy —3 zs 
queftion, fo oN 2 » KC. - - -- eee to ee are whole numbers. Put ~==/3 
12>) ao 9 2 TI 
: ‘ oe.) Tes  ebttiie B29 
whence y==11f ; which fubftituted for y in the fecond term, gives =wh, and aa 
wh. which put =r, whence p—ror-+g, and y=110r-tg9. By proceeding in like manner with 
all the terms to the laft, we fhall have y=277200-++-2 5:9; and by taking v==0, T, 2; 33 4, 5> by 
7, 8,9, 10, and 11, we fhall have refpeétively the numbers 2519; 30239, 57959 85679» 
(113399; 1411193 £68339, 196559, 224279, 251999, 279719, 3074393 Which numbers will 
anfwer the fecond condition of the queftion.—V.B. In this folution zo/ ftands-for whole number. 
The fame anfwered by Mr. R. Simpson, of Bath. 
The leaft common multiple of 2, 3,4, 5, 6, 7, 8, 9, and 10, is 2520(—=10X 2X 9X 7) if this- 
number be therefore increafed by unity, the fum will evidently be a number which, divided by 
‘LO, 9, 8, &c. would leave the given remainder 1. Hence the expreffion 2520x-+-1 will detez~ 
mine any quantity of fuch numbers, by taking x==3, 2,3, &c, But, per queftion, thefe num-= 
bers ought to be divifible by 11; to include this condition, place 11 as a denominator to the above 
2520x-L ; xt 
expreflion, viz. 2520x-F1 and the quotient a » Or its equal nage tt _mutt be 

an integer, and confequently, an integer alfo. Affume it =z, and x is found =117—<1 ; 
pies 
which determines the firft part of the queftion: for, by expounding x by 1, 2,3, &c. and fubfti-- 
tuting the correfponding values of x in the expreflion above, any feries of numbers may be found 
having the required properties, the leaft twelve of them being 25201, 52921, 80641, 108361, 
336031, 163801, 191521, 219243, 246961, 274681, 302401, and 330121, thofe re- 
quired. 
; The feeond part of the queftion is anfwered in the very fame manner, only inflead of incréaf- 
ing we muft diminifh the Ieaft commen multiple by unity, and then the refulting value of x will 
be rim-+y. Let 2==1, 2, 3, &c. and the refpeétive values of x fubftituted in this expreffion 
2520x——1 will'give fo many numbers fulfilling the conditions in the fecond part of the queftion : 
the required twelve are, 2519; 30239, 57959, 35679, 113399, I411I9, 368839, 1985593 
224279) 251999) 279719» 3°7439- 
NEW QUESTION XXIII. By Mr. R. Simpson, of Bath, 
Given the bafe, and the ratio of the fides; to conftruct the triangle, when the fupplement of 
one angle at the bafe is equal to the complement of the other. 



HALF-YEARLY RETROSPECT 
"i OF THE STATE OF 
DOMESTIC LITERATUR Ra es 
~ 
qi ig a circumftance which has a pro- 
mifing afpeét on the future ftate of 
fociety, that the attention of the literary 
public is gradually turning from unpro- 
ductive fpeculations, to fubjeéis of gene- 
ral utility.. The workd is growing tired 
of thofe abftraét refearches, which lie 
remote from the great ends of human 
exiftence, and is wifely empioying itielf, 
principaliy, in thofe enquiries which tend 
~ to correét the diforders of fociety, and to 
meliorate the condition cof man. Many 
of the publications which are to come 
under our confideration in our -prefent 
periodical retrofpeét, are cf this ufeful 
kind, particularly in the clafs which we 
fhall firf{ introduce to the notice of our 
readers. ‘ 
POLITICAL ECONOMY. 
Upon that branch of Pelitical Economy, 
which every confifient friend of human 
kind muft acknowledge to be the moft 
interefting fubjeét which can, in the pre- 
fent fiate of fociety, occupy the thoughts 
of the philofopher, er engage the exer- 
tions of the’ ftatefman, Sir PREDERIC 
EDEN has beftowed laudable induftry,. 
in his large and valuable work, entitled 
‘« The State of the Poor ;.or a Hiftory 
of the Labouring Claffes in England, 
from the Congueft to the prefent Period.” 
~Theugh we are inclined to think the 
fiatements in this work refpeéting po- 
pulation, inaccurate, and the conclufion 
in favour of the prefent time, erreneous ; 
and though we have no doubt, that a 
com parifon 
” 
