Text: Chapter One Concerning the definition of Logarithms and nomenclature Logarithms are numbers which keep equal differences with adjoining proportional ones.1 For any whatsoever of these given numbers, the other numbers, and these being different from those that are able to be adjoining [in proportion], which with no trouble agree with the general definition of Logarithms, and they will be able to give a not unwelcome benefit for these others. Thus, if the numbers in continuing proportion were: 1, 2, 4, 8, 16, 32, 64, 128, ... The numbers labelled A, B, C, or D can be adjoining these [numbers and act] as logarithms, as you see here [Table 1-1]. Or for others, with this one method being kept: as, together increasing or decreasing, the differences of the logarithms shall be equal, as often as the numbers to which they have been adjoined are proportional.2 For not inconveniently, Logarithms may be called: equally differing companions of numbers in proportion. That they seem to be called logarithms by their most distinguished inventor [John Napier] for this reason: because they reveal to us numbers keeping the same ratio between themselves.3 Furthermore, so that we can reach a better understanding of the nature of these Logarithms and of our disposition towards them, certain Lemmas should be considered. First Lemma For any numbers whatever being set up equally increasing or decreasing, the differences of the same with the intervals of the same are proportional.4 Let the first, third, and eighth numbers from column D be taken: 35, 29, 14 : between the first and third are two intervals; between the third and eighth are five intervals. I assert, that the first to third difference 6, to be as the third to the eighth difference 15, as 2 to 5. Therefore with a sequence of numbers in continued proportion, with any two logarithms whatever given, we will be able to find the logarithm of any other [given number].5 For being given the interval between these numbers themselves, and the other [interval] from the third [number] and either of the two [numbers], as well as the difference of the given Logarithms. Therefore, with two intervals and the difference of the given logarithms being given, with the other [interval] being given, the difference of the logarithms sought will be found by proportionality [ and hence the other logarithm]. For let 4, 6, 9, 13 1/2 , 20 1/4 , 30 3/8 , be the numbers in continued proportion, and let the logarithms of the first and third numbers being given, namely 060206, and 095424, the difference of which is 035218. The logarithm of the number 30 3/8 is required, namely the sixth of those given. The given interval between the first and the third is two, between the third and the sixth three; the given difference of their logarithms 035218, the proportional fourth number sought 052827 is the difference of the logarithms sought, which on adding to the logarithm of the third number, gives 148251 as the logarithm of the required sixth number. As this [Table 1-2] shows:6 Second Lemma If from four numbers, the first quantity is in excess of the second, just as much as the third exceeds the fourth: then the sum of the first and forth will be equal to the sum of the second and third, and conversely. As for 9, 5, 15, 11, so the sum of the middle numbers, as of the extremes, is 20. See Proposition 4, Book 1, of Bachet's Porisms on Diophantus, [the 1621 edition].7 And these two lemmas, on logarithms in general, showing quite well their especial characteristics. Notes On Chapter One 1 If a and b, c and d are any two pairs of adjoining positive numbers in proportion: a/b = c/d, then the difference of the 'logarithms' associated with a and b is the same as the corresponding difference of the logarithms associated with c and d. 2 For the numbers in proportion form a Geometric Progression, while their powers form an Arithmetic Progression; there are thus endless possibilities for the choice of numbers used. See note 4 below. 3 An interesting article in the Mathematical Gazette for 1934, p. 192 - 205, John Napier, by W. R. Thomas, explores the possible origin of the word, and also sheds some light on the life of John Napier. 4 These equally increasing or decreasing numbers form a AP, and so represent possible logarithms. As may be easily shown, all AP's with a general term of the form {A + B cross n} obey the same rules of proportion as the simplest sequence {n}. Hence, we are to think of the logs as members of an AP progressing uniformly with each 'application' of the ratio [the members of the GP themselves being in continued proportion]. This is of course a trivial exercise using a base and indices, but these tools were not available at the time. 5 Let the three numbers be a, b, and c; with log a and log b known, and log c to be found. The interval between values of the ratio is proportional to the difference of the logarithms of the numbers, according to note 4 above. As both the intervals b - a and c - b (or c - a if we wish) are known, and the difference of the logs of the first interval is known, then the difference of the logs of the second interval can be found, and subsequently the log of the third number c. 6 Briggs uses an arbitrary AP for possible logarithms: 7, 9, 11,13,15,17, ... to find the given ratio, as well as using his base 10 tables. 7 This Lemma follows immediately from the definition of logs at the start of this chapter. While this definition gives no hint as to how the logarithms are to be constructed by Briggs, is a useful starter in their use given a set of Briggs' tables of logarithms. http://www-groups.dcs.st-and.ac.uk/~history/Miscellaneous/Briggs/Chapters/Ch1.html

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