1878.] Dr Pearson, On a Manuscript Table of Log. Sines. 147 



80", 0' to 30". 5'. is printed below. The natural sines, &c., are of 

 course nearly identical with those now used ; but the logarithms 

 are the original logarithms actually invented by Napier, and in their 

 form, though not otherwise, differ from the natural or hyperbolic 

 logarithms, generally so called ; and which have been published 

 at length by Wolfram and Dase. They decrease instead of in- 

 crease ; and those given in this table are nearly the same as those 

 printed by Wright, at London, in 1618, and by Ursinus, at 

 Cologne, in 1624 ; though much more full, as will be seen by com- 

 parison. Thus, according to Wright, the log, sine of 0" 1' is 

 8-142567, that of 0" 2', 7-449421... while that of 89" 30' is 

 •000038-1 ; and that of 89" 59' only -OOOOOO'l. Mr Leigh's manu- 

 script, however, only commences with 5" ; there is a gap from 

 82" to 33" 59', and consequently also from 56" to 57" 59' inclusive, 

 and it ends with 85". 



The imperfect state of the tables, and the time at which the pro- 

 bable author lived, compared with the date at which the much more 

 convenient tables to the base 10 were issued by Briggs and Vlacq 

 (viz. 1624, 1628 for numbers, and 1633 for the Trigonometrical 

 functions), lead us to suppose that if written out by Mr Leigh in his 

 younger days, they were, at any rate, not thought by him worthy 

 of completion ; and it is not unlikely that he may have obtained 

 them from some of his predecessors. 



It may be well to insert here an example of the method in 

 which Prof. De Morgan (Eng. Cyc. Art. Tables of Logarithms) 

 shows that logarithms, such as are given in these tables, may 

 be converted into natural, and also into modern logarithms. For 

 example, we have the natural sine of 30" given as '500000, but its 

 log. is -693147. Subtracting this from 6907755, which is the natural 

 logarithm of 1000, we have as remainder 6-214608. Dividing this 

 by the modulus 2-30258 we get very nearly 2-69897, viz. the com- 

 mon logarithm of .500. Subtracting from this the common log. of 

 1000, viz. 3, we have 9-69897 as the common log, sin. of 30". 



Vice versa, the natural sine of 5" is about '087. Now the 

 common log. of 87 is 1-9395193, and multiplied by the modulus, 

 this becomes 4-465908, and this again subtracted from 6-907755 

 (the natural log. of 1000) leaves as remainder 2-441747. In the 

 tables we find 2449058, a sufficiently near approximation, but it is 

 to be observed that neither Wright nor this MS. insert any dot or 

 comma after the first 2, though it will be seen by following the 

 tables through that, on the system now adopted, it would be 

 required. 



took his M.A. degree at Oxford, and in 1664 the degree of S.T.B. at Cambridge. 

 He went afterwards to Bishop Stortford, and niiich information respecting him 

 will be found in Chauncy's History of Herts. The library which he formed for 

 the benefit of the school is still in existence. 



Vol. III. Pt. iv, 11 



