152 
NATURE 
[ FEBRUARY 3, 1923 

Greek Geometry, with Special Reference to Infinitesimals.! 
By Sir tf. i, Hears; KCB KC WeOren, Ros: 
(te geometry passed through several stages 
from its inception to its highest development 
in the hands of Archimedes and Apollonius of Perga. 
The geometry which Thales brought from Egypt early 
in the sixth century B.c. was little more than a few 
more or less accurate rules for the mensuration of 
simple figures ; it was the Greeks who first conceived 
the idea of making geometry a science in and for 
itself. With Pythagoras and the Pythagoreans, who 
represent the next stage after Thales, geometry 
became a subject of liberal education. Apart from 
special discoveries such as those of the theorem of 
the square on the hypotenuse, the equality of the 
three angles of any triangle to two right angles, the 
construction of the five cosmic figures (the five regular 
solids), and the incommensurability of the diagonal 
of a square with its side, the Pythagoreans invented 
two methods which remained fundamental in Greek 
geometry, that of proportions (though in a numerical 
sense only) and that known as application of areas, 
which is the geometrical equivalent of the solution 
of a quadratic equation. 
By about the middle of the fifth century the Pytha- 
goreans had systematised the portion of the Elements 
corresponding to Euclid Books I., II., IV., VI., and 
probably III. 
In the second half of the fifth century, concurrently 
with the further evolution of the Elements, the Greeks 
attacked three problems in higher geometry, the 
squaring of the circle, the trisection of any angle, 
and the duplication of the cube. Hippias of Elis 
first trisected any angle by means of his curve, the 
quadratrix, afterwards used to square the circle. 
Hippocrates of Chios, who also wrote the first book 
of Elements and a treatise on the squaring of certain 
lunes, reduced the problem of duplicating the cube 
to that of finding two mean proportionals in continued 
proportion between two straight lines, the first solu- 
tion of which was by Archytas, who used a wonderful 
construction in three dimensions. Democritus, among 
many other mathematical works, wrote on irrationals ; 
he was also on the track of infinitesimals, and was the 
first to state the volume of any pyramid and of a cone. 
The fourth century saw the body of the Elements 
completed. Eudoxus discovered the great theory of 
proportion set forth in Euclid Book V. and the “ method 
of exhaustion” for measuring curvilinear areas and 
solids. Theztetus contributed to the content of 
Book X. (on irrationals) and Book XIII. (on the five 
regular solids). This brings us to Euclid (fl. about 
300 B.C.). 
To the third century B.c. belong Aristarchus of 
Samos, who anticipated Copernicus ; ‘and Archimedes, 
who, with Apollonius following after twenty years 
or so, concludes the golden age of Greek geometry. 
To come to the history of infinitesimals. The 
Pythagoreans discovered the incommensurable and 
maintained the divisibility of mathematical magni- 
tudes ad infinitum. The difficulties to which the 
latter doctrine gave rise were brought out with in- 
1 aedayes from the presidential address to the Mathematical Association, 
January 2 
NO. 2779, VOL. 111] 
comparable force by Zeno in his famous Paradoxes 
and in other like arguments. Zeno’s Dilemmas pro- 
foundly affected the lines on which mathematical 
investigations developed. Antiphon the Sophist, in 
connexion with attempts to square the circle, declared 
that by inscribing successive regular polygons in a 
circle, beginning with a triangle or square and con- 
tinually doubling the number of sides, we shall ulti- 
mately arrive at a polygon the perimeter of which 
will coincide with that of the circle. Warned by Zeno’s 
strictures, mathematicians denied this and substituted 
the statement that by following the procedure we can 
draw an inscribed polygon differing in area from the 
circle by as little as we please. Similarly they would 
never speak of the infinitely great or infinitely small ; 
they limited themselves to postulating that by con- 
tinued division of a magnitude we shall ultimately 
arrive at a magnitude smaller than any assignable 
magnitude of the same kind, and that by continual 
multiplication of any magnitude however small we 
can obtain a magnitude exceeding any assignable 
magnitude of the same kind however great. On this 
safe basis Eudoxus founded the whole of his theory 
of proportion and the method of exhaustion. 
It has been remarked that the method of exhaustion, 
though a conclusive method of proof, requires previous 
knowledge of the result to be proved, and is of no 
use for discovering new results. This is scarcely 
true because, before the proof by veductio ad absurdum 
is applied, the area or volume has to be exhausted, and 
this process often indicates the result. The process 
means a summation of a series of terms; and there 
are different classes of cases according to the nature 
of the series to be summed. In one case (Archimedes’ 
quadrature of a parabolic segment) the summation 
is that of the geometrical progression 1 +}+(4)?+. : 
Archimedes sums this, nominally, to m terms ‘only. 
He says nothing about taking the limit when x is 
increased indefinitely, but merely declares that the 
area of the segment, Rs is actually A{r+4+(4)? 
+ ... ad inf.}, is 4 A, where A is the area of a 
certain triangle. It seems plain, nevertheless, that 
Archimedes found his result by mentally taking the 
limit. Other series summed by him are arithmetical 
progressions and the series of the squares of the 
first m natural numbers. In these cases Archimedes 
sums two series representing respectively figures 
‘circumscribed and inscribed to the figure to_ be™ 
measured, and then states a certain intermediate 
quantity to be the actual area or content required. 
Here again Archimedes, though he does not say so, 
states what is in fact the common limit of the two 
sums when the number of terms in the series is in- 
definitely increased, and a factor common to all the 
terms is at the same time indefinitely diminished. — 
The result is actually equivalent to integration. There 
are some six cases of the kind depending on the in- 
tegrations /xdx, [x%dx, [(ax+x*)dx and f sin 6d@ taken 
between proper limits respectively. 
The reasons why the Greeks were so limited in the 
number of integrations which they could directly 
effect were that they had no algebraical notation and 

