\ 
Vor. IV.] Mathematical Correfpondence, ‘ 5sg 
term 4/c—d. But, it may be afked, what fignification have thefe lat terms? I anfwer, that 
the term 4/—a fignifies the fquare root of —a, or in other words, the fquare root of a negative 
quantity, and is what mathematicians call an impoffible er imaginary quantity. (See Maclaurin’s 
Algebra, Part I. chap. viii. fet. 4g; and Part IL. chap. i. fect. 8.) The other term fignifies the 
{quare root of the difference of the quantities c and 4. : 
Let us now, Mr. Editor, take a view of the progrefs we have made. We have difcovered 
that Philo-Cofa’s affertions are not generally true ; that they are true only when the terms are 
independently confidered, in which cafe they may be made: to fignify any thing at all, no matter 
what ; and laftly, that they are all of them fale, when applied to thofe terms as they occur in 
equations. Now, as it was profefledly in this light that I confidered them, wiz. as they really 
occur in equations, I think it will follow of courfe that Philo-Cofa’s affertions, and confequent 
reafoning on them, will fall to the ground. 
After this deduction, it may feem unneceffary to-take any farther notice of Philo-Cofa’s oba 
jections ; yet, left any one fhould think that his argument againft the Corollary, as he has been 
pleafed to call it, ought to have been difproved, I will here briefly confider it. To avoid cas 
villing, I will grant him as far as /—2x Yl VED _ “© Thus,”’ fays he, “ the fecond 
power of the 4/ —a is not —a, but ba.” Againft this conclufion I thus argue: 4/—a x 
s/—4=-+4 3 confequently, by evolution, 4/ —a =/ +a, i.e. an imaginary or impoffible 
quantity, equal to a real pofitive one, which is abfurd ; therefore his conclufion is falfe. 
Having now, I prefume, Mr. Editor, done away all Philo-Cofa’s objections againft my paper, 
1 would beg leave to obferve, that the definition is not neceffary to the exiftence of the ftruéture, 
bat only tends, as I think, to make the fubjeét more iptelligible. The ftructure will ftand 
without it. With each cafe is given, what appears to me to be the only fubitantial illuftration of 
its truth of which it feems capable. And if thefe cafes, upon every occafion in real practice, 
give true refults, furely every uleful purpofe is anfwered. 
The fubject of negative and imaginary quantities is by no means a difficult one of itfelf, It 
can be confidered in only two puints of view: frffy as it relates to equations ; /econd/y, in the 
abftraét, or independently. Confidered in the firft point of view, there can be but one opinion 
concerning it: it is in this light only that the fubjeé can be at all ufeful 3 it is in this light that 
the illuftrious Newton has confidered it. Confidered in the fecond point of view, the terms 
may be made to fignify any thing or nothing, at the caprice of the ufer: it is in this light that 
the terms feem to admit of an indefinite number of fignifications, each of which may be true 
as here confidered, but falfe when applied to real ufe. 
Let us now fee, Mr. Editor, if we have not difcovered the funken rock on which mathema- 
ticians have foundered. 
They confider te terns indefendently : in this light their conclufions are ‘true 3 but when thefe 
eonclufions are applied to the fame terms as they occur in equations, is it any wonder that they 
fhould be falfe > This is the rock on which Mr. Emerfon has- foundered. when confiderins the 
quantities mentioned in the remark at page 117 of this Magazine. Ibis upon a corner of the ~ 
fame rock that my good friend Philo-Cofa_has fplit. 7 
If any of your ingenious correfpondents fhould think it neceflary to make any farther rernarks 
en this fubjeét, I could wifh that they would confider it ferioufly ; it furely deferves {uch a 
confideration ; mathematical truths are not to be ridiculed and laughed out of countenance. 
After thanking you, Mr. Editor, for the indulgence you have granted me, believe me to be 
Your obliged fervant, 4 ' 
Newcaftle-upon-Tyne, . J. Garnett. 
OF. 14th, 1797. 
ee 
To the Editor of ihe Monthly Magazine. 
SIR, 
AGREE in opinion with your ingenious correfpondent, Philo-Ccfa, fo far as he has confidered 
the doétrine of Imaginary Quantities But as Mr. Garnet has, ’n his fecond and third cafes, 
drawn conclufions different from thofe of al! other writers upon the fubjeé&t * (and which have 
net been noticed in the reply to his paper) here felicit indulzence to examine their reétitude. 
. . = xr 
Mr, Garnet has, in his third cafe, endeavoured to prove, from the equation am — —= ¢, that 
; Vv 
the value of the produét of the imaginaries 4/—a, ./—b will give the refult + 4/a ; although ~ 
he has before determined (fee cafe r) that when thefe factors are {uppofed equal, the refult - 
BAe ER 
* Mr, Garnet has miftaken Profeffor Euler’s conclufions ; they are each determinately +L, ox 
each determinately —. See Profeffor Hutton’s Dictionary, under the article «€ Imaginary 
Quantities.” 
Mentuty Mac: XXVI, 4D would 
