sbct. i.] DISSERTATION SECOND. 37 



the use of the signs, and profit by a simplification, which 

 is the work entirely of the algebraick language, and cannot 

 be imitated by any other. 



That I might not interrupt the view of improvements so 

 closely connected with one another, I have passed over 

 one of the discoveries, which does the greatest honour to 

 the seventeenth century, and which took place near the 

 beginning of it. 



As the accuracy of astronomical observation had been 

 continually advancing, it was necessary (hat the correct- 

 ness of trigonometrical calculation, and of course its diffi- 

 culty, should advance in the same proportion. The signs 

 and tangents of angles could not be expressed with suffi- 

 cient correctness without decimal fractions, expending to five 

 or six places below unity, and when to three such numbers 

 a fourth proportional was to be found, the work of multipli- 

 cation and division became extremely laborious. Accord- 

 ingly, in the end of the sixteenth century, the time and 

 labour consumed in such calculations had become exces- 

 sive, and were felt as extremely burdensome by the mathe- 

 maticians and astronomers all over Europe. Napier of 

 Merchiston, whose mind seems to have been peculiarly 

 turned to arithmetical researches, and who was also devoted 

 to the study of astronomy, had early sought for the means of 

 relieving himself and others from this difficulty. He had 

 viewed the subject in a variety of lights, and a number of 

 ingenious devices had occurred to him, by which the te- 

 diousness of arithmetical operations might, more or less com- 

 pletely, be avoided. In the course of these attempts, he 

 did not fail to observe, that whenever the numbers to be 

 multiplied or divided were terms of a geometrical progres- 

 sion, the product or the quotient must also be a term of 

 that progression, and must occupy a place in it pointed 

 out by the places of the given numbers, so that it might be 



5 



