196 
History. struct a system for himself, guided by the recollec- 
tion of the conversations which he had heard among 
the mathematicians that visited his father, who was 
himself a mathematician. He had gone as far as to 
discover that the thrée angles of any triangle were equal 
to two right angles, when he was observed by his fa- 
ther. At the age of 16, he is said to have composed a 
treatise on conic sections, in which all that Apollonius 
had demonstrated was elegantly deduced from a single 
proposition: this was shewn to Descartes, but the phi- 
losoper could not believe it to be the work of so young 
a geometer. The hopes he had so early excited, and 
the elegance of his disquisitions on the cycloid, gave 
geometers reason to regret that a larger portion of his 
short life was not dedicated to the science. He died in 
1662, aged 39. 
Gregory St Vincent, a Flemish mathematician, held 
a respectable place among the geometers of his day. 
The main object of his researches was the quadrature 
of the circle, which he sought with the most persever- 
ing industry —— all the difficulties of the geome- 
try of his time. He even believed he had succeeded ; 
but in this he was wrong: his researches, however, 
procured him a rich harvest of other geometrical 
truths. 
Andrew Tacquet; another Flemish mathematician, 
was a respectable geometer. He endeavoured to ex- 
tend the boundaries of the science by a treatise on the 
mensuration of the surface and solidity of bodies form- 
ed by cutting a cylinder in different ways by a plane, 
and of different solids formed by the revolution of seg- 
ments of circles and conic’ sections. In treating of 
these, he has affected the rigorous style of the ancient 
demonstration, a thing not entitled to commendation, 
considering that it was by adopting a more brief style, 
and new views, that the science was then receiving 
great improvement. 
The celebrated Huygens was one of the brightest 
ornaments of that period. At an early age, he pub- 
lished his Theoremata de Circuli et hyp. quad. He 
completed what Snellius had done concerning approxi- 
mations to the circle, in his work De Circule Magnitu- 
dine invenia ; these were the labours of his youth: af- 
terwards he found the surface of conoids and spheroids, 
a problem which, on account of its difficulty, had not 
been attempted before his time. He determined the 
measure of the cissoid ; he shewed how to reduce the 
problem of the rectification of curve lines to that of 
quadratures ; and he invented the theory of involutes 
and evolutes. His treatise De Horologio Oscillatorio, 
is the finest wy canes that has ever been given of the 
application of the most profound to mecha- 
nics. In short, his name is associated in the history of 
geometry with some of the most brilliant discoveries 
that have been made in the science. 
Our countryman, James Gregory, also stands in the 
very highest class as a geometer. He treated of the 
quadrature of the circle, and gave better methods of 
approximating to it than were known before his time. 
e attempted to shew that the complete solution of the 
sna was a thing impossible ; but the correctness 
is reasoning was questioned by’ Hw . In 1668, 
Gregory published Tis Goometris pary Onivetii 
which gave the first idea of the logarithmic curve, and 
contained many curious theorems useful for the trans- 
formation and quadrature of curvilineal figures, for the 
rectification of curves, and for the measure of their so- 
lids of revolution, &c. He wrote various other works, 
some of which belong rather to the modern analysis 
Gregory St 
Vincent. 
Born 1584. 
Died 1667. 
Tacquet. 
Died 1660. 
Huygens, 
Born 1629. 
Died 1695. 
J, Gregory. 
Born 1632. 
Died 1675. 
GEOMETRY. 
than to the ancient geometry. The excellence of his 
writings, and their rareness, has induced Mr Baron 
Maseres to reprint tliem at his sole expence, as a testi- 
mony of his estimation of the author’s merit, and to 
make the elegance of his views, and the extent of his 
claims as a discoverer, better-known. Our mathema- 
tical readers will readily recollect, that this is not the 
only obligation of the kind that this worthy man has 
conferred upon science. See Grecory. ' 
Dr Barrow next claims our attention by his admira- garow, 
ble geometrical writings; his geometrical lectures are Born 163 
composed partly in the style of the ancient, and partly Died 16 
in that of the modern geometry. He had the high ho- - 
nour of being the geometrical tutor of Newton, to 
whom he resigned his mathematical professorship, 
with a view to dedicate his time to theological stu- 
dies ; but seduced from his purpose by his favourite 
science, he did homage to it, by giving an edition of 
the writings of Archimedes, Apollonius, and Theodo.« 
sius. Such was this excellent man’s estimation of geo- 
metry, that he considered the contemplation of it as 
not unworthy of the Deity. The beginning of his 
Apollonius was inscribed with the words, @¢os yeoseress, 
Tu autem Domine ree es geometra, “ God himself 
2 
geometrizes; O Lord, how great a — thou art!” 
In Italy, Torricelli, the disciple Galileo, cultivated To 
geometry: with such a master, it is easy to conceive Born 160 
any degree of excellence in the scholar. Among other Pied 1647, 
geometrical enquiries, he treated of the solid formed by 
the rotation of ah about its RE pp vee 
he shewed that it had a finite magnitude, a thing which 
may appear paradoxical, when it is considered that the 
generating surface is infinitely great. 
Borelli also claims attention on account of his edi- Borelli. 
tions of Euclid, A ius, and Archimedes, works re- Born 1608. 
markable for their brevity and perspicuity ; and also Pied 167% 
because of his efforts in ing from the Arabic - 
three books of the Conics of Apollonius, which were 
then supposed to have been lost. See Conic Src 
TIONS. , . , 
Viviani, another disciple of Galileo, must here also Viviani. 
be noticed. His geometrical writings were of the most Born 1622: 
elegant and valuable kind.. We have spoken, in our Died 1703: 
treatise on Conic Sections, of his restoration of the 
Conics of Apollonius ; and in our treatise on FLux1uns, 
(art. 165.) of his beautiful problem concerning the 
“a 
quadrature of a ion of the surface of a sphere. 0 
We have already noticed some of Descartes’s geos 
metrieal labours, but his main effort, for which his 
name will be handed down to posterity with honour, 
was his application of algebra to geometry ; an inven- "3 
tion by which the properties. of figures 
were represented by equations. is Geometry, which ’ 
contains his views on this subject, was published ‘first 
in 1637. The union of geometry and promo- 
ted very much the discovery of the new calculus, the 
germ af which lay concealed in the method of exhaus- 
tions of the ancients, was partly evolved by Cavallerius, 
and still farther in the arithmetic of infinites of Dr 
Wallis, and, lastly, fully expanded by Newton and 
Leibnitz. The history of geometry becomes now in-— 
terwoven with that of the modern analysis, and is — 
chiefly interesting by the extent to which the science 
has been carried by that powerful instrument of inven- - 
tion. by doLel ey yur 
Although the ancient geom was thus in’a Mat- yp cion, 
ner supplanted by the nt. Pegs or the seience by Mat G4 
no means lost its interest. Sir Isaac Newton held it Died 
in such esteem, that he delivered his sublime discove- 
