The History of Mathematical Proof in Ancient Traditions

494 agathe keller

Th e standard formulation of the Rule of Th ree, applied to the

similar triangles AFD and DEC, can be recognized here. Th e standard

expression of the Rule of Th ree provides the proportional elements on

which the computation is based. Here the rule indicates that the ratio of

AF to FD is equal to the ratio of DE to EC. Th e ‘reinterpretation’ of the

rule thus gives the arbitrary set of operations a mathematical signifi cance.

Rather than just a list of operations, the rule in Ab.2.15 becomes a Rule of

Th ree.

2.3.2 A gnomon in relation to the celestial sphere

In the previous commented verse (BAB.2.14), Bhāskara sets out two proce-

dures. Both rest on the proportionality of the right-angled triangle formed

by the gnomon and its midday shadow with the right-angled triangle

composed by the Rsine of the altitude and the zenithal distance. In the

present example, one procedure uses only the Rule of Th ree, while the other

uses the Rule of Th ree with the Pythagorean Th eorem. Both procedures

compute the same results.

Consider Figure 14.3. Here, GO represents a gnomon and OC indicates

its midday shadow. Th e circle of radius OSu (Su symbolizing the sun) rep-

resents the celestial meridian. Th e radius OSu is thus equal to the radius

of the celestial sphere. S′u designates the projection of the sun onto the

horizon. Th e segment SuS′u illustrates the Rsine of altitude. Bhāskara I

notes that the triangle SuS′uO is similar to GOC. Th erefore the segment

S′uO (that is, the Rsine of the zenithal distance) is proportional to the

shadow of the gnomon at noon and the Rsine of the altitude is propor-

tional to the length of the gnomon. Th is proportionality is further illus-

trated in Figure 14.4.

In modern algebraic notation,

SuS u

GO

SuO

OC

SuO

GC

′

=

′

=^

Th e mathematical key to this situation is the relationship between the

celestial sphere and the plane occupied by the gnomon, which Bhāskara

and Āryabhat. a call ‘one’s own circle’ ( svavr. tta ). Th is relationship is high-

lighted here by a set of puns. Th us, the gnomon and the Rsine of the alti-

tude have the same name ( śa ̇nku ), as do the shadow of the gnomon and

the Rsine of zenith distance ( chāya ). GC is the ‘half-diameter of one’s own

circle’.