THE GENESIS OF LOGARITHMS 155 



which contains the germ of Napier's great discovery. We 

 append a literal translation of the passage in question : 



" It is also of some use to know this property. If a series 

 of numbers be arranged in a geometrical progression from 

 unity and any two of the terms of that progression be multiplied 

 together, the product will also be a term in the same pro- 

 gression ; and its place will be at the same distance from the 

 larger of the two factors that the lesser factor is from unity ; 

 and its distance from unity will be the same, minus one, that the 

 sum of the distances of the two factors from unity is distant 

 from unity. For, let A, B, C, D, E, F, G, H, /, K, L represent 

 any geometrical progression from unity, of which A is the 

 unity ; let D be multiplied by H, and let X represent the 

 product. Take L in the given progression, which is at the 

 same distance from H that D is from unity. It is to be demon- 

 strated that X is equal to Z,." 



This proposition Archimedes proceeds to prove, giving also 

 the proof of the second proposition quoted in the above trans- 

 lation. Now this amounts to neither more nor less than 

 demonstrating that, given a geometrical progression, the pro- 

 duct of any two terms can be found without going through the 

 actual process of multiplication. The following would be 

 an equivalent method of stating the second of the above 

 propositions : 



Take any geometrical progression starting from unity and 

 underneath each term write its " distance from unity," placing a 

 o underneath unity. Thus : 



1, 2, 4, 8, 16, 32, 64, 128 .. . 

 0123456 7 . . . 



Then, to multiply 4 by 16, we add 2 and 4 together and look 

 up the number (64) above 6, which gives the required result. 

 It is to be noted that by starting the lower progression at o, 

 we get rid of the " minus one " of the proposition as quoted by 

 Archimedes. 



But this — the study of the relation between an arithmetical 

 and a geometrical progression — is precisely the manner in which 

 the problem was approached by Napier. And his great insight 

 is shown, both in the manner in which he obtained a pro- 

 gression or series of geometrical progressions such that the 

 terms of the series were very near in value to the numbers in a 

 table of natural sines — for it is to be remembered that primarily 

 Napier was seeking for a table of logarithms of sines — and by 



