In 1717 Abraham Sharp published in his Geometry Improv'd the Briggian logarithms of numbers from 1 to 100, and of primes from 100 to 1100, to 61 places; these were copied into the later editions of Sherwin and other works.
To Napier seems to be due the first use of the decimal point in arithmetic. Decimal fractions were first introduced by Stevinus in his tract La Disme, published in 1585, but he used cumbrous exponents (numbers enclosed in circles) to distinguish the different denominations, primes, seconds, thirds, &c. Thus, for example, he would have written 123.456 as 123@4050603.
(1) he saw that a point or separatrix was quite enough to separate integers from decimals, and that no signs to indicate primes, seconds, &c., were required; (2) he used ciphers after the decimal point and preceding the first significant figure; and (3) he had no objection to a decimal standing by itself without any integer.
In the old times the grand-prince was simply primes inter pares among the minor princes, and these lived with their boyars almost on a footing of equality.
The seventh article provided that bountied sugars (sucres primes) must be excluded from import into the territories of the signatory powers, by absolute prohibition of entry or by levying thereon a special duty in excess of the amount of the bounties, from which duty sugars coming from the contracting countries, and not bountyfed, must be free.
For square measure 12 square inches = I superficial prime, 12 superficial primes = I square foot; while for cubic measure 12 cubic inches = I solid second, 12 solid seconds = I solid prime, 12 solid primes = I cubic foot.
When an area has been calculated in terms of square feet, primes and square inches, the primes and square inches have to be reduced to square inches; and similarly with the calculation of volumes.
His best-known papers, however, deal with prime numbers; in one of these (" Sur les nombres premiers," 1850) he established the existence of limits within which must be comprised the sum of the logarithms of the primes inferior to a given number.
His table gives the logarithms of all numbers up to 2200, and of primes (and also of a great many composite numbers) from 2200 to 10,009, to 48 decimal places.
His method of finding the logarithms of the small primes, which consists in taking a great number of continued geometric means between unity and the given primes, may be described as follows.
As has been stated, Abraham Sharp's table contains 61-decimal 10 b= log 24 = - log (1-160) d =10g 49 = - log (1-160) 17253 8 35 62 21868 Briggian logarithms of primes up to I ioo, so that the logarithms of all composite numbers whose greatest prime factor does not exceed this number may be found by simple addition; and Wolfram's table gives 48-decimal hyperbolic logarithms of primes up to 10,009.
All the great rivals of Abdalmalik having now disappeared, he was no longer like his predecessors primes inter pares, but dominus.
Factors, Primes and Prime Factors.
Thus 2, 3, 5, 7 and I I are primes, for each of these occurs twice only in the table.
The number r is usually included amongst the primes; but, if this is done, the last paragraph requires modification, since 144 could be expressed as 1.2 4.32, or as 1 2.2 4.3 2, or as IP. 2 4.32, where p might be anything.