12 SIR J. COCKLE ON MATHEMATICAL HISTORY. 



5. Lagrange {ib. p. 20) says that it does not appear that 

 the ancients knew o£ the law of virtual velocities. From 

 what is said by Whewell in his ' History of the Inductive 

 Sciences' (1837^ vol. ii. pp. 39 and 42, vol. i. pp. 69, 81^ 

 82, and 94) it seems that the law^ so far as it relates to 

 the lever, was known to Aristotle. Whewell considered 

 the physical philosophy of Aristotle {ib. i. 25) and of the 

 Greeks {ib. p. 33) an utter failure. He apparently re- 

 garded the doctrine of the lever (see vol. ii. p. 59, and 

 vol. i. chap. i. sect, i, p. 91) with greater favour than that of 

 virtual velocities (see vol. ii. chap. ii. sect. 4, p. 39), in 

 connexion with which he makes no special mention of 

 Lagrange (see vol. ii. p. 120). Professor Cayley (^Mes- 

 senger/ N. S., vol. iii. p. i) has given a dissertation " On 

 the General Equation of Virtual Velocities.''' 



6. There are two modes of approaching the parallelogram 

 of forces. First, we may, from the composition of motions, 

 through the second law of motion, pass to the composition 

 of forces. Thomson and Tait, fortified by the example of 

 Newton (see their 'Treatise' &c., vol. i. pp. 181, 182, 

 § 255-257), have followed this mode, which they believe 

 to contain the most philosophical foundation for statics. 

 Lagrange, whose >vritings are a mine of historical infor- 

 mation, says (Mec. Anal. i. 13) that the ancients knew 

 the composition of motions, as we see by some passages of 

 Aristotle in his ' Mechanical Questions,' and that the 

 geometers especially have employed it for the description 

 of curves, as Archimedes for the spiral, Nicomedes for the 

 conchoid, &c. 



7. Secondly, we may seek to arrive at the composition 

 of pressures, independently of the second law of motion, by 

 processes which are valid whether that law be a law of 

 nature or not, and which would be valid even if we had not 

 any conception of motion, and which, indeed, do not render 



