SINES AND VERSED SINES TRIGONOMETRY. 309 



minute ; and this is all the exactness aimed at in their trig- 

 onometrical tables. The author, however, does not mean to 

 assert that the ratio of the radius to the circumference is 

 either accurately, or even very nearly, as 3438 to 21,600, 

 which makes the diameter to the circumference as 1 to 

 3.14136. It appears from the Institutes of Akbar that the 

 Bramins knew the ratio of the diameter to the circum- 

 ference to greater exactness, and supposed it to be that of 1 

 to 3.1416. " 



The tables employed in their trigonometrical calculations 

 are two, — one of sines, and the other of versed sines. The 

 sine of an arc they call cro.majya or jyapinda, and the versed 

 sine utcramajya. These temis seem to be derived from the 

 word jya, which signifies the chord of an arc, from which 

 the name of the radius or sine of 90°, viz. trijya, is also 

 taken. This regularity in their trigonometrical language is 

 not unworthj' of remark ; but what is of more consequence 

 to be observed is, that the use of sines, as it was unknown 

 to the Greeks, who calculated by the help of the chords, 

 forms a striking difference between theirs and the Indian 

 trigonometry. It is generally supposed that the use of sines, 

 instead of chords, in modern trigonometr)', was borrowed 

 from the Arabians. It is certainly one of the acquisitions 

 which the mathematical sciences made when, on their ex- 

 pulsion from Europe, they took refuge in the East. 



The table of sines exhibits them to every twent3--fourth 

 part of the quadrant ; the table of versed sines does the 

 same : in each the sine or versed sine is expressed in 

 minutes of the circumference, neglecting fractions. Thus, 

 the sine of 3° 45' is 225, the sine of 7° 30' is 449, and so 

 on. The rule for the computation of the sines is curious ; 

 it indicates a method of computing a table by means of 

 their second differences, — a considerable refinement in cal- 

 culation, and first practised by the English mathematician 

 Brigjs. 



The Surya Siddhanta does not give the demonstration 

 of the truth of the rule ; but the commentary gives direct 

 geometrical means for their calculation. In the progress of 

 science, the invention of trigonometry is a step of great 

 importance, and of considerable difficulty. He who first 

 formed the idea of exhibiting in arithmetical tables the 

 ratio of the sides and angles of all possible triangles must 



