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SCIENCE PROGRESS 



appended, Napier calls the "Radical Table"; a specimen of 

 the part of this table given in the Constructio is shown below : l 



The Radical Table 



With the help of this Radical Table it is an easy matter to 

 obtain the logarithm of any given sine, as, given the sine, the 

 number nearest to it in the Radical Table must be noted and by 

 the " difference theorem " quoted above the difference between 

 the logarithms of the two numbers may be found. Adding 

 this difference to or subtracting it from the logarithm of the 

 number in the Radical Table at once gives the logarithm of the 

 given sine. 



From this Radical Table, therefore, the logarithms of the sines 

 of all angles between 90 and 30 are computed. Further than 

 this we cannot go, without other assistance, as the natural 

 numbers in the Radical Table only go down to (about) half-radius, 

 which is the sine of 30°. It remains, then, to explain the methods 

 adopted by Napier in computing the logarithms of the sines of 

 the angles between 30 and o°. 



Two methods are indicated by means of which the com- 

 putation may be effected. In the first, 2 the given sine .r, which, 

 by hypothesis, is the sine of some angle less than 30 , is 

 multiplied by some definite number b, the number b being so 

 chosen that the product b x (=y, say) lies within the limits of the 

 Radical Table. This being so, the number nearest toy is looked 

 up in the Radical Table and by the " difference theorem " earlier 

 quoted the logarithm of y may be evaluated. Then, knowing 

 the logarithms of y and of b, the value of the logarithm of x is 

 obtained from the equation jy = b x. 



In the second method, 3 Napier utilises the proposition that 

 *' As half-radius is to the sine of half a given arc, so is the sine 



1 Construction Prop. XLVII. 2 Ibid. Prop. LI.-LIV. 3 Ibid. Prop. LV. 



