290 Mathematical Correfpondence. : (April, 
To the Editor of the Monthly Magazine. 
SIR, 
READING an abftraét of Euler’s Algebra, by Profeffor Esert, of Wittenberg, in the Ger- 
man language, I was aftonifhed to find, that the new principle, given by Mr. Lesite, in 
the fecen: vol. of the Edin. Phil. Tranfactions, for the folution of indeterminate problems, and 
noticed, at p. 633, vol. 1. of Dr. Hur ton’s Math. and Phil. Dict. is borrowed from that work, 
as weli as fome other things containedin the fame paper. 
‘The principle, as given by Mr. Lesir£, is this: 
If A XB be any compound quantity == C XD, and m be any rational number affumed at 
pleafure, then will AXmB=C XmD, and tuppofing AmD, it will follow, that wB—=C or 
B==—. And thus we obtain two equations of a lower dimenfion,.and, by affuming, multiples 
92 ~ 
mand /, if thefe equations be capable of farther decorapefition, form four equations ftill more 
fimple. By therepeated application of this principle, an higher equation, admitting of divifors, 
will be tefolved into thofe of the firft order, the number of which will be one greater than that 
of the multiples affumed. 
Now Mr. EuLer makes ufe of it in this form (fee his folution of the cafe ax7--5=0, &e.) 
If, as before AXB=CXD, multiply both fides of the equatien by #7, and AXBXfge=CX 
Djg .. if Af=Cq, Bg Dg or A=ce and ‘Bebe. but thefe values by putting ma are mae 
- 
‘ 
nifeftly the fame as Mr. Leflie’s, viz. A=wC, and B=—, and needs to have nothing farther 
op : : 
faid to eftablifh it ; though, to exemplify ita little, we thall jufttake Mr. Leflic’s fir examples 
which is this: 
To find two rational numbers, the difference of the fquares of which may be a given 
number ? 
Let the given number ==ab, and put x and y for the two rsequiredones. Then is x7=—-y?= 
x—y. aly = ab, multiply each fide by p7, and it is weeny, x-by. pa==ab XL9 5 fuppofe ay, 
— a 
q==bp and xy. f==ag the firft divided by g gives gaye and the latter by /, say 

g 
‘ , b bk b sas 
thefe added together, give a Or x== sated and yD etl or fubftituting » fo 

47 (2f “af 
hey b PET ona yt 1d EEE sehen re te ea 
the ecome 4==— —. — = ang y== = _. ee, precifely tne tame as F 
y “oe 2 2m at 2m = om P + ee 
Mr. Leflie. 
What farther may be obferved, is this: that Mr. Leflie’s folution of the general quadratic 
Ax?+Bx-+-C= 0 js to the very fame effect as Euler’s, and each making thefe four cafes of it : 
ift. When A= (, or the expreffion is of the form 2x7-+Brt-C=D. 
ad. When C=, or the expreffion of the form A?C*-Ba-+C*+—-O. Ne 
3d, When neither A nor C are fquares, but the expreffion capable of being divided into two 
fa&ors, which it will be whenever B-—4AC=D. 
ath. When the general quadratic can be divided into parts, fuch that the one fhall be a fquare, 
anathe aig capable of being refolved into factors, which will produce g form of the kind 
an live = 
a 
o enter ; ee eT. 
eae as alretay been faid, no doubt can remain of the fact, 
sey lensth: for after what has alretay 2S ; 
nec ee | . I am, fir, your’s, &c. es 
Neweaftle-upon-Lyne, Feb. 11, 1797: cei 
P.S. Problem VIII. of Mr. Lestre’s examples, wherein he propofes, from having eerie 
values ¢ and d,for « and _y in the expreflion ax?-|-b=y* to determine others, is likewile taken 
from the fame work. 
ee 3 = > Tey sell] - 
‘ato a minute COmpe: fon of particulars, would only be fwelling my paper to an un 

QUESTION. XXXI. 
itl 
Require to place two equal ftraight lines at fuch an angle a eee mayen 2: best ene 
. circle is defcribed about the angular point, and the extremities of the be we. amok Shae 
of the fides of an equiangular and equilateral pentagony igi i in it. = mie ae 
for exagon, Heptagon, Octagon, Nonagon, Decagon, Wudeeapne oO g 2 ates 
gecagon, without recourfe to the central angles of any of the aforefaid hguies, nor to g 
thofe figures. e ¥ick 
ORIGINA 
