GEOMETRY. 
he published his divination with the title Apollonius Ba- 
tavus. He also treated of the approximate value of the 
circumference of a circle in his Cyclometria. He here 
shewed how Van Ceulen might have greatly shortened 
his labour by two limits nearer to the circumference 
than the circumscribing and inscribed polygons ; and 
he verified the calculation, by computing the perimeter 
of a polygon of 1073741824 sides, which, according to 
the other method, would have given only 20 figures of 
the number. . : 
Albert Girard, another Fleming, was highly esti- 
eter, He was the first that found the 
rical triangle, or of a polygon’ bound- 
ed by great circles on the sphere. He deserves still 
more honour, however, for his divination of the Porisms 
of Euclid, if, as he asserted, he really had succeeded in 
restoring the work of the ancient geometer. Unfortu- 
nately his labours on this subject have never been 
published. 
Want of room obliges us to pass over several whose 
reputation as geometers is excelled by that which they 
acquired in other branches of mathematics: we must 
not, however, omit the celebrated Kepler ; he was the 
ote ee 
* the idea of in into anguage of geometry. e 
circle he considered as composed of an infinite number 
of triangles, having their vertices at the centre, and of 
sich toe bases form the circumference ; and the cone, 
as made up of an infinite number of pyramids, whose 
bases formed its base, and which had with it a common 
__ vertex. 
. By the aid of these, and similar views, Kepler, in 
his Nova Slereometria, a work on gauging, demonstra- 
' ted, im a direct manner, and with great brevity, those 
truths which the cone had ne by irene 
very peculiar | es of reasoning. er open 
ae book a vast field for speculation ; for, passing 
beyond the views of Archimedes, he formed a multi- 
tude of new bodies, and he investigated their solidi- 
ties. Archimedes limited his enquiries to those gene- 
rated by the rotation of conic sections about an axis, 
but Kepler treated of solids generated by the rotation 
these curves about any line whatever in their plane. 
He thus considered ninety solids besides those handled 
by the Sicilian geometer. Upon the whole, this book 
contained views which appear to have had great influ- 
ence on the improvements that soon afterwards took 
place in, geometry. 
The problems proposed by Kepler probably Jed to 
the invention of the methods of Guldin and Cavalle- 
- rius.. The principal discovery of Guldin consisted in 
an application which he made of a rty of the 
eentre of gravity, to the measure of solids produced by 
revolution.‘ Every figure,” says he, “ formed by the 
rotation of a line, or a surface about an immoveable axis, 
is'the product of the generating quantity by the line 
by its centre of gravity. . This principle, cer- 
tainly one of the most beautiful discoveries in geome- 
try, was however known in the days.of Pappus ; for 
it is distinetly stated at the end of the pretace to his 
seventh book ; yet Guldin takes no notice of the cir- 
cumstance, ie ; 
_ To Cavallerius we are indebted for the doctrine of 
indivisibles, which heypublished in 1635. In this, he 
considered a line as made up of an infinite number of 
_ Points, a surface, -of an infinite number of lines, and a 
solid, of an infinite number of surfaces: these elements 
of itudes he called Jndivisibles, The introduction 
of so bold a postulate’ into geometry, was opposed by 
some of his contemporaries ; but in answer, the Italian 
‘discoveries in geometry 
195 
geometer explained that this hypothesis was by no 
means an essential part of his theory, which, in fact, 
was the same as the ancient method of exhaustions, 
but free from its tedious and indirect modes of reason- 
ing. 
4 the first place, he considered such figures as 
had their increasing or decreasing elements at equal 
heights above the base, always in a given ratio; and 
he shewed that the figures themselves were to each 
other in the same given ratio. Next, he compared 
figures composed of an increasing or decreasing se- 
ries of elements, with others in which the elements 
were all equal: for example, a cone, which he con- 
sidered as composed of an’ infinite number of circles, 
increasing from the base to the vertex, with a cylinder, 
which is composed of an infinite number of circles, all 
of the same size ; and to determine the ratio of the con- 
tents of the two solids, he found the ratio of the sum 
of the decreasing circles in the cone, to the sum of the 
circles which were equal to one another in the cylinder. 
In the cone, these circles decrease from the base to the 
vertex as the squares of the terms of an arithmetical 
progression. In other solids, they form other progres- 
sions; for example, in the parabolic conoid, it is.simply 
that of an arithmetical progression. The general.object 
of the method is to assign the ratio of this sum of an in- 
creasing or decreasing’ series of terms with that of the 
equal terms which form’an uniform and known figure 
of the same base and altitude. The method of indivi- 
sibles is now superseded by the more extensive doc- 
trine of fluxions; yet it was of immense importance 
at the time it Was invented, and in fact was one step 
towards that grand discovery. 
The French geometers pursued the same career of 
discovery, and almost at the same time as Cavallerius ; 
they even resolved more difficult problems. In 1636, 
Fermat had found the area of a spiral, ofa different 
nature from that which Archimedes had handled ; and 
soon afterwards, he proposed to Roberval to determine 
the areas of parabolic curves of the higher orders, (See 
Fermat.) Roberval quickly resolved the problem ; and 
he also determined their tangents. Fermat, again, on his 
part, found their centres of gravity. Roberval claimed 
the merit of having invented for himself a theory altoge- 
ther similar to that of Cavallerius, before the latter had 
~made his known ; but as his selfish views led him to 
conceal it, that he might triumph over his contempora- 
ries, he has but little claim on the gratitude of: poste- 
rity as a discoverer, although he deserves credit for his 
skill asa sae: Roberval ven venture to deyi- 
ate so much from the common of geometry as 
Cavallerius ; he-conceived his sp tert solids nid be 
made up of an ‘indefinite number of very narrow rect- 
angles and thin prisms, which decreased according to a 
certain law. i 
‘The celebrated Deseartes contributed in:no-small de- 
History, 
—_—ve 
Fermat. 
Died 1665. 
Robervai. 
Born 1602, 
Died 1675. 
Descartes. 
gree to the developement of these new and brilliant Born 1596. 
When Mersennus had sent 
him an account of Fermat’s method of finding the centre 
of gravity of conoids, Descartes quickly sent him the 
determination of the centres of gravity of all parabolas, 
also their general quatltature, their tangents, and the 
ratios of their conoids. 
It was in this period that the logarithmic spiral and 
cyeloid were brought into discussion ; the former was 
suggested by Descartes, the latter was first noticed by 
Galileo, See Ericycxor. 
Died 1650. 
» Passing over many geometers of ordinary merit, we Pascal. 
must, notice’Pascal,;» who, at the age. of twelve, had 
such a turn for geometry, that he undertook to con- 
Born 1623. 
Died 1662, 
