20. 
For the Mouthly Magazine. 
A SKETCH of the HISTORY Of PURE MA- 
THEMATICS, tranflated from “ Traiié 
Elementaire de Mathematiques Pures, 
par LuMoiNne, Profefeur de Maibema- 
tigueset de-Phyfique, Oc. 
. ALGEBRA. 
[ Continued from page 527, Vel. xiii.] 
HE reification of a curve line, 
firft peformed. by Neil, was only 
an extenfion of the new views which Wal- 
lis had opened in his Arithmetic of Infi- 
nites, a work pregnant with genius, the 
appearance of which may be regarded as 
an epocha in the progrefs of modern geo- 
metry: Guided by the thread of analogy, 
of which he always knew how to avail 
himfelf, the Englifh analyft obferved, 
that the denominators of fraétions may be 
confidered as powers with negative expo- 
nents. ‘This remark enabled Wallis to 
meaiure all the fpaces, whofe elemen's 
are inverfely as fome powers of the ab- 
{ciffz. 
$0. We alfo owe to Wallis the method 
known by the name of interpolation, 
which confifts in inferting, in a progref- 
fion of quantities, proceeding by a cer- 
tain law, oneor more intermediate terms, 
conformable to that law. In endeavour- 
ing to interpolate, in a certain progref- 
fion, a term which he expected wou!d give 
him the area of the circle, Wailis only 
found an infinite feries of terms, which 
converted more and more to the true 
value. Not being fatisfied with this re- 
fult, he invited Lord Brounker* to fecond 
his efforts; and his Lordfhip, by means 
of continued fraciions, of which he was 
the inventor, gave a more juft approxi- 
mation. 
81. To Brovnker, geometry is alfo in- 
debted for having firkt expreffed the area 
of the hyperbola in an infinite feries. 
Mercator,+ who had difcovered a fimilar 
feries, publifhed that fine difcovery in his 
79° 
* Lord Brounker, Vifcount of Caftle Lions 
in Ireland, was born about the year 1620, 
and died in 1684. When the Royal Society 
was firft eftablithed,; he was elected Prefi- 
dent, and was annualiy continued tn that 
office fora period of about fifteen years. 
+ Nicholas Mercator was a native of the 
Dutchy of Holfein, but went to England 
about the year :660, and refided in that 
country till his death. He was one of the 
firf{ Members of the Royal Society. Theat 
celebrated mathematician ‘was {aie to have 
been fo weak, as to helieve in judicial aftro- 
logy. 
Sketch of the Hiftary of Pure Adathematics. 
[ Aug. I, 
Logarithmotechnia, which was printed ir 
the year 1668. , 
$2. Barrow,* the cotemporary of Wal- 
lis, publifhed his Geometrical Leétures 
in 1669. Among the excellent imven- 
tions, which he has explained in that 
work, filled as it is with profound im- 
quiries, we ought particularly to obferve 
his method of drawing tangents to curves. 
That Englifh Geometrician confdered the 
* Ifaac Barrow was born -in London about 
the year 1630. | His youth juftified no great 
expectations; but having arrived at manhood, 
he made a rapid progrefs in all kinds of know- 
ledge. His merit procured him the Greelk 
profefforfhip in Cambridge, which he‘ ex-— 
changed for that of Geometry. Barrow died 
in 1678. 
Note by the Tranflator. —Inftead of any bio- 
graphical addition to our author’s hints re- 
ipeGing Dr. Barrow, the reader may perhaps 
be pleafed to fee a tranflation of a curious 
paflage in the preface which that confum- 
mate geometrician, that preceptor and pre- 
curfer of Newton, has prefixed to his edition 
of Appollonius, publifhed in 1675: This 
paflage appears to me to poffefs all the fim- 
plicity and fublimity, with very little of the 
quaintnefs, cbfervable in the fingular Dedi- 
dication of Edwards’s Hifory of Birds: - 
7) Ore 
“ Tz autem, Domine, quantus es Geometra,” &&. 
*¢ But how great a geometrician art thou, 
PT oe ti) a 
For this fcience knows no limits, and even 
human fagecity can difcover numberlefs new 
truths: but Thou perceiveft them all at ane 
view, without any chain of dedu€tions, or 
tirefome length of demonftrations. In other 
fubjects, our intelleét poffefies but little 
power: like the imagination of brutes, it 
feems only to dream of fome uncertain ob- 
jets, concerning which there are almof as 
many opinions as there are men. Butin ma- 
thematical truths, there isan univerfal agree- 
ment 5 in them the human mind feems capable 
of fomething great and wonderful, &c.—Thee, 
therefore, 1 rejoice to love; to ihee I look 
up, ardently longing for that day, when thy 
immenfe and mof holy benignity fhall enable 
me to underftand, not only thefe, but far 
more numerous and impoitant truths, with 
a mind purged from error and prejudice, and 
without this fucceffive and laborious effort of 
thought.”—-Barrow in his Oratio prafatoria, 
on being placed in the chair of geometry, dif- 
yewmeteet—GOD adfs geometrically. 
‘plays the vaft utility of the mathematical 
fciences more elegantly and forcibly than 
any author 1 know. But that piece, though 
brief for the fubje&t, would be too long for 
this place. The original may be feen im his 
LeGiones Mathem. Cantab. habita, A.D. 1664, 
&c. and a tolerable tranflation in Stene’s Ma- 
thematical Dictionary, article Méathemaiiis. 
a little, 
