THE GENESIS OF LOGARITHMS 153 



that of the tangents is obtained from the theorem " As cosine 

 is to sine, so is radius to tangent " — the equivalent of tan 6 — 



„ — whilst, using theorems such as " The secant of an arc 



cos 6 & 



is equal to the sum of its tangent and the tangent of half its 



complement," or " The secant of an arc is equal to the difference 



between the tangent of that arc and the tangent of the arc 



added to half its complement," the canon of secants is deduced 



from that of tangents, by simple additions and subtractions. 



Such, then, was the state and efficiency of the trigono- 

 metrical tables known to the mathematical world at the begin- 

 ning of the seventeenth century. The labour involved in such 

 computations as those that we have detailed above as well as 

 the increasing accuracy of astronomical observations gave rise 

 to a demand for a method of calculation which should materially 

 lessen such labours. That method was given to the world by 

 John Napier in the invention of logarithms. These aids to 

 calculation are looked upon as so much a matter of course at 

 the present day and are so strongly associated with " powers," 

 "indices" and what not, that the curious mode in which they 

 originated is apt to be lost sight of. In the writer's view this 

 is unfortunate. Nothing is more common than to hear and 

 read discussions as to whether the modern schoolboy shall be 

 taught to handle logarithmic tables before he is taught the 

 theory of indices and the subsequent deduction of logarithms 

 or not— some holding the latter view, others asserting that to 

 tell a boy he shall not use a table of logarithms until he knows 

 the theory of their construction is as inconsequent as to forbid 

 a boy's using his watch until he knows how to make one of 

 those useful articles. Yet the fact is never brought forward 

 that the discoverer of logarithms had not the ghost of a notion 

 of an index as we know it and that complete tables of logarithms, 

 the direct ancestors of those we use to-day, were printed and 

 in daily use close on a century before the days of Euler, who 

 was one of the first, if not the first, to look upon logarithms 

 as being indices of powers. 



Of the life of the discoverer of logarithms few details are 

 known. Born in 1550 and living a life of retirement in a 

 country which was notably wild and lawless even in a lawless 

 age, the antiquary will find little that will help him to recon- 

 struct the daily life of John Napier. A great mass of Napier's 



