
1799: | 
are poflible or not; and whether 2, 6, 10, 
&c. or 0, 4, 8, &c. are impoflible, from 
the laft term of it being poflible or more ; 
this rule was generally given in 1762, 
and the latter obfervation inferted in the 
Philofophical Tranfactions for the year 
1764.—A rule is given for finding the 
number of impoflible roots, from deducing 
an equation, whofe roots are the refults of 
the roots of the equation, commonly called 
the equation of limits, fubftituted in the 
given equation, &c.; this finds all that 
the preceding method does, and equally 
with that finds the true number of im- 
poflible roots contained in any equations 
of 3, 4, or 5 dimenfions; and perhaps 
more generally finds the true number of 
impoflible roots than any rule yet given: 
it may eafily be rendered more general. 
I fhall mention the following rules, be- 
caufe they have been fince publifhed by 
eminent Mathematicians.—1, A rule for 
finding impoffible roots from the equation 
whofe roots are the fquares of the roots 
of the given equation, &c. 2. The finding 
the number of aflirmative, negative and 
iinpoffible roots in an equation, whofe 
roots bear any aflignable relation to the 
roots of the given equation, from the 
numbers of them in the given equation. 
3. In the common refolutions of cubic 
and biquadratic equations, by the differ- 
ent roots of the given equations, are ex- 
prefled the roots of the reducing one: 
and wice verfa from them are difcovered, 
how many roots of the given one are im- 
poffible, &c. There are given more rules 
for finding the number o7 impoffible roots 
In equation, containing one or more un-, 
known quantities,one of which always dil- 
covers them, when Sir Ifaac Newton’s rule 
does, and oftentimes when it does not. 
Two elegant theorems are given for finding 
when fome poffible quantities are neced- 
farily greater than others; thefe may be 
demonitrated, by proving their difference 
to be the aggregate of certain {quares. 
There are fome theorems, which give the 
ratio of the contents of all the quantities 
refulting from fubftituting the roots of 
one equation, for the unknown quantity 
in the other refpeétively, and multiplying 
their refults: this is inferted, becauie 
feveral elegant properties of parabolas 
have been deduced from it—-Some truths 
are deduced concevning equations, of 
which the roots are the limits of each other. 
In my preface to the algebra, the num- 
ber.of inventions enumerated in thefe two 
chapters are 9 and 19 refpectively, their 
fum 28; imeach of the three fucceeding 
ehapters, the number enumerated is more 
MonruLy Mac. No. Xbiv, 
Original Letter of the late Dr. Waring. 
3°9 
than the abovementioned fum 28. The 
number of inventions contained in the 
Meditationes Analytice, on the Modern 
Analyfis, Fluxions, Seriefes, &c. is many 
more than the number in theAlgebra; and 
the number of properties of curve lines 
deduced is not greatly lefs; to thefe add 
the number in the addenda, Phulofophical 
Tranfactions, and tranflation of algebraie 
equations into propable relations, and the 
number will be more than four hundred. 
It would be too much trouble to review 
fuch a number of propofitions, particular- 
ly as they are for the moft part enumerated 
in the prefaces to the books themvelves. 
I might equally particularife my in- 
ventions in all other branches of mathe- 
matics: in properties of algebraical 
curves and folids, in conic lettions, &c. 
there are more new properties,and feveral of 
them, as appears to me, of the firit degree 
of elegance,contained perhaps in them than 
in the works of any other writer, and in 
many the algebra and principles from 
which they are deduced, were alfo invented 
by me. Since the publication of the book, 
I have given in the addenda feveral new 
properties, and extended moft of the pro- 
perties of circles of Archimedes and 
Pappus to conic fections, and rendered 
fome more general, and given in the Phile- 
fophical Tranfaétions arule for the demon- 
{tration of fevera] propoftions contained 
in them and fimilar ones, from principles: 
of algebra. 
Many new feries are derived from dif- 
ferent principles; fome of whtch are the 
moft converging, but in thefe cafes it is 
commonly neceffary that a near approxi- 
mate fhould be given, which will be the 
firft term of the feries.—Rules are given. 
for finding the convergency of feries, and 
for rendering them converging ae they 
exprefs the fluents of fluxions, contained 
between different values of the variable 
quantities ; new integrals of increments ; 
fums of feries,of which the terms are given. 
From approximations to the different 
roots, or, to two or more of algebraical | 
equations, areadduced more near approxi- 
mations.—Something is added of the 
difficulties which occur in finding the 
feries for the fluents of fluxional equations. 
Therule of falle is rendered more general, 
by finding nearer approximations, when 
two or more approximations are given, 
and the errors of their refults! The fame 
is applied to more unknown quantivies 
contained ; anew method of differences 
and correspondent values is added, with 
fome problems thence deduced, &c. 
Many new propofitions are invented, 
r in 
ee 
