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22 THE NATURE OF MATHEMATICS 

the heavens, for the purpose of map-making, we have to 
measure angles for example, by turning a sight, like those 
used on guns, through an angle measured on a circular arc of 
metal to fix the relative directions of the stars or points on 
the earth. Now, for terrestrial measurements, a piece of 
country is approximately a flat surface, while the heavens are 
surveyed as if the stars were, as they seem to be, scattered on 
the inside of a sphere at whose centre we are. Secondly, it is 
a network of triangles plane or spherical of which we 
measure the angles and sometimes the sides : for, if the angles 
of a triangle are known, the proportionality of the sides is 
known ; and this proportionality cannot be concluded from 
a knowledge of the angles of a rectangle, say. Hipparchus 
(born about 160 B.C.) seems to have invented this practical 
science of the complete measurement of triangles from certain 
data, or, as it is called, " trigonometry," and the principles laid 
down by him were worked out by Ptolemy of Alexandria (died 
168 A.D.) and also by the Hindoos and Arabians. Usually, 
only angles can be measured with accuracy, and so the question 
arises : given the magnitude of the angles, what can be con- 
cluded as to the kind of proportionality of the sides. Think 
of a circle described round the centre O, and let AP be the 
arc of this circle which measures the angle AOP. Notice that 
the ratio of AP to the radius is the same for the angle AOP 
whatever value the radius may have. Draw PM perpendicular 
to OA. Then the figure OPMAP reminds one of a stretched 
bow, and hence are derived the names " sine of the arc AP " 
for the line PM, and " cosine " for OM. Tables of sines and 
cosines of arcs (or of angles, since the arc fixes the angle if the 
radius is known) were drawn up, and thus the sides PM and 
OM could be found in terms of the radius, when the arc was 
known. It is evident that this contains the essentials for the 
finding of the proportions of the sides of plane triangles. Spheri- 
cal trigonometry contains more complicated relations which are 
directly relevant to the position of an astronomer and his 
measurements. 

Mathematics did not progress in the hands of the Romans : 
perhaps the genius of this people was too practical. Still, it 
was through Rome that mathematics came into medieval