22 REPORT—1883. 
ninth and tenth books fully and ably developed the first principles of 
the Theory of Numbers, including the theory of incommensurables. 
We have next Apollonius (about B.c. 247), and Archimedes (b.c. 287- 
212), both geometers of the highest merit, and the latter of them the 
founder of the science of statics (including therein hydrostatics) : his 
dictum about the lever, his ‘ Evpy«a,’ and the story of the defence of 
Syracuse, are well known. Following these we have a worthy series of 
names, including the astronomers Hipparchus (B.c. 150) and Ptolemy 
(A.D. 125), and ending, say, with Pappus (A.D. 400), but continued by their 
Arabian commentators, and the Italian and other Huropean geometers of 
the sixteenth century and later, who pursued the Greek geometry. 
The Greek arithmetic was, from the want of a proper notation, 
singularly cumbrous and difficult; and it was for astronomical purposes 
superseded by the sexagesimal arithmetic, attributed to Ptolemy, but 
probably known before his time. The use of the present so-called Arabic 
figures became general among Arabian writers on arithmetic and astro- 
nomy about the middle of the tenth century, but was not introduced into 
Europe until about two centuries later. Algebra among the Greeks is 
represented almost exclusively by the treatise of Diophantus (A.D. 150), in 
fact a work on the Theory of Numbers containing questions relating to 
square and cube numbers, and other properties of numbers, with their 
solutions ; this has no historical connection with the later algebra, intro- 
duced into Italy from the East by Leonardi Bonacci of Pisa (a.p. 1202- 
1208) and successfully cultivated in the fifteenth and sixteenth centuries 
by Lucas Paciolus, or de Burgo, Tartaglia, Cardan, and Ferrari. Later 
on, we have Vieta (1540-1603), Harriot, already referred to, Wallis, 
and others. 
Astronomy is of course intimately connected with geometry ; the most 
simple facts of observation of the heavenly bodies can only be stated in 
geometrical language: for instance, that the stars describe circles about 
the pole-star, or that the different positions of the sun among the fixed 
stars in the course of the year form a circle. For astronomical calcula- 
tions it was found necessary to determine the arc of a circle by means of 
its chord: the notion is as old as Hipparchus, a work of whom is 
referred to as consisting of twelve books on the chords of circular ares ; 
we have (A.D. 125) Ptolemy’s ‘ Almagest,’ the first book of which contains 
a table of arcs and chords with the method of construction; and among 
other theorems on the subject he gives there the theorem afterwards 
inserted in Euclid (Book VI. Prop. D) relating to the rectangle contained 
by the diagonals of a quadrilateral inscribed in a circle. The Arabians 
made the improvement of using in place of the chord of an arc the sine, 
or half-chord of double the arc; and so brought the theory into the form 
in which it is used in modern trigonometry: the before-mentioned 
theorem of Ptolemy, or rather a particular case of it, translated into the 
notation of sines, gives the expression for the sine of the sum of two arcs 
in terms of the sines and cosines of the component ares; and it is thus the 
