5267. 
68. The analytic art remained for fome 
time in the ftate. in which Vieta left it. 
The firft fteps in its farther improvement 
were made by Harriot*, who fimplified the 
notation of the French analyft, by intro- 
ducing the ufe of {mall letters initead of 
capitals. Harriot firft thought of carrying 
all the terms of an equation to~ one fide, 
fo as to make the whole exoreflion equal 
to nothing; and hence every value, pofi- 
tive or negative, which, when fubftituted 
for the unknown quantity and its powers, 
in fuch an equation, makes it equal to 
nothing, will be the value, or one of the 
values, of that’ unknown quantity. In 
fine, Harrtot made the important obferva- 
tion, that all equations of many dimen- 
fions may be confidered as the produéts of 
equations of one dimenfion. This man- 
ner of viewing the generation of equations 
makes it evident, that, in every equation, 
the unknown quantity hath as many va- 
ues as thereare units in the index in its 
higheft and characterifing dimenfion. 
69. Such are nearly all the difcoveries 
svith which Harriot enriched analyfis. But 
no geometrician has contributed fo much 
to its progrefs as Defcartes. He firft 
fhewed how to mark with numerical expo- 
nents the powers to which the fame letter 
arifes by repeated multiplications. To 
him we owe our knowledge of the nature 
and ufe of negative roots. He taught us 
that thofe roots, which were rejeéted as 
ulciefs: by preceding analyfts, are as real 
as pofitive roots; that, like them, they 
ferve to folve a problem, and that be- 
tween pofitive and negative roots there is 
no difference but inthe manner of confi- 
dering the quantities which they repre- 
fent, 
70. Defcartes is alfo the author of rhat 
fine rule, which, in an equation all whofe 
roots are real, fhews us how to determine, 
by the bare infpe&tion of the figns, the 
number of pofitive and of negative roots 
belonging to that equation. He is alfo 
the inventor of the method of ufing in- 
determinate co-efficients, which is of great 
importance, and much uled, in the theory 
* Thomas Harriot was born at Oxford in 
1§60, and died in 1621. The work in which 
he colleéted his own analytical difcoveries 
and thofe of his predeceffors, is intitled Arcis 
Analytza Praxis. From this book, Wallis 
pretends that Defcartes, whom he treats as a 
plagiary, copied what he wrote on analyfis. 
But the Englith are the only people who re- 
fufe the honour of the invention to Defcartes, 
in order to afcribe it to their own country- 
man.—See the tranilator’s remarks at ihe ede 
Sketch of the Hiftory of Pure Mathematics. 
[July L 
of equations, and in a great number of 
mathematical problems. 
71- Some mathematicians before Def- 
cartes had applied algebra to geometry, fo 
that he had only the merit of extending 
the ufe of that method. But the honour 
of having applied algebra to the theory of 
curves is wholly his own, He confidersa 
curve as formed by the extremities of va- 
riable lines, which have certain propor- 
tions to other variable lines ; and the ex- 
preffion of thofe proportions in algebraic 
language prefented to him a table, in 
which he read, {a to {peak, all the aifections 
of the curves which he confidered. 
7z. But of all the difcoveries of Def- 
cartes, that which gave him the mott fatis- 
faétion, and appeared to him the mot ules 
ful, was his general rule for drawing tan- 
gents to curves. ‘To determine the tan- 
gents, he has left us two ingenious me- 
theds, founded on the fame principle. 
73. Betore Defcartes pubiifhed his Ge- 
ometry, Fermat was in poffcflion of a me- 
thod of determining the Maxima and Mi- 
zima. By means of this method, it was 
ealy to draw tangents to curves, by confi- 
dering a tangentas a fecant, and by mak- 
ing the interval between the ordinates, 
corre{ponding to the points of interfeétion, 
to vanifh. From hence there rémained 
but one ftep to the Differential Calculus ; 
but Fermat was not the man who made it. 
We-may obferve, by the way, that Fer- 
mat made deep difcoveries in the theory 
of prime numbers, and that he inveftigated 
feveral fine properties of figurate num- 
bers, while Pafcal was penetrating déeply 
into their nature by means of his arithme- 
tical triangle, gibt 
74. The Geometry of Defcartes did 
not meet with.an univerfal good reception, 
Roberval ftrove to deprefs it; but De 
Baume*, Schootent+, Huddet, Van Hez- 
raet§, &c. {upported it with zeal, and en- 
deavoured to difplay all its merit. 
* Florimone de Beaune, Counfellor or 
Prefidial of Blois, was born in 1601, and died 
in 1651. Defcartes had fuch a friendfhip for 
him, that, in feveral of his letters, he de- 
clares, that he relied more on his Jeariing 
and approbation, thanon thofe of all the other 
geometricians then in France. 
+ Francis Schooten was a Profefior at Ley~ 
den. ) 
t John Hudde, a Burgomaiter of Amfter- 
dam, died at a great age in 1704. Having 
been as great in politics as in geometry, he’ 
ferved his country in diftinguifhed fituations, 
and contributed, by his difcoveries, to the ad- 
vancement of the {ciences. 
§ Van Heuraet was a Dutchman. 
| Wore 
