8), the continual mention of large decimal numbers of shekels in the earlier books, and the certain fact of 100 shekels being = mina.
In these cases the digit itself, or decimal multiples, seem to have been used.
The first national coinage was begun in 1822, and the decimal system was adopted in 1863.
The great spread of the Phoenician weight on the Mediterranean, of the Persian in Asia Minor and of the Assyrian in Egypt are evident cases; and that the decimal weights of the laws of Manu (43) are decidedly not Assyrian or Persian, but on exactly the Phoenician standard, is a curious evidence of trade by water and not overland.
The former translated the work into English; the latter was concerned with Napier in the change of the logarithms from those originally invented to decimal or common logarithms, and it is to him that the original calculation of the logarithmic tables now in use is mainly due.
Which stand as (1/6)th of the other decimal series based on the digit.
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.
A decimal system of numeration was used, with numbers going up to io,000.
And a strange division of the shekel in 10 (probably therefore connected with this decimal mina) is shown by a series of bronze weights (44) with four curved sides and marked with circles (British Museum, place unknown), which may be Romano-Gallic, averaging 125/10.
The power of io, which occurs as a factor in the tables of both Napier and Byrgius, was rendered necessary by the fact that the decimal point was not yet in use.
The Greek system, however, adopted this foot as a basis for decimal multiplication, forming --
The explanation of this property of the base io is evident, for a change in the position of the decimal points amounts to multiplication or division by some power of 10, and this corresponds to the addition or subtraction of some integer in the case of the logarithm, the mantissa therefore remaining intact.
An account has now been given of Napier's invention and its publication, the transition to decimal logarithms, the calculation of the tables by Briggs, Vlacq and Gunter, as well as of the claims of Byrgius and the method of prosthaphaeresis.
He also reduced the solar parallax to 14" (less than a quarter of Kepler's estimate), corrected the sun's semi-diameter to 15' 45", recommended decimal notation, and was the first to make tidal observations.
These examples show that Napier was in possession of all the conventions and attributes that enable the decimal point to complete so symmetrically our system of notation, viz.
The accuracy of a meter is tested by drawing calibration curves showing the percentage departure from absolute accuracy in its reading for various decimal fractions of full load.
Seeing the good reasons for this digit having been exported to the West from Egypt--from the presence of the 18.23 cubit in Egypt, and from the 0.729 digit being the decimal base of the Greek long measures--it is not surprising to find it in use in Italy as a digit, and multiplied by 16 as a foot.
3 Besides his connexion with logarithms and improvements in the method of prosthaphaeresis, Byrgius has a share in the invention of decimal fractions.
In the Swedish House of Nobles his contributions to political discussion had great influence, and he dealt with such subjects as the currency, the decimal system, the balance of trade and the liquor laws (where he was the pioneer of the Gothenburg system) with marked ability.
According to Krummel the following relations hold, good at 18° C. for the monochromatic light of the D line of the sodium spectrum in units of the fifth decimal place.
It originated in Babylonia as the foot of that system (24), in accordance with the sexary system applied to the early decimal division of the cubit.
It is remarkable how near this early decimal system of Germany and Britain is the double of the modern decimal metric system.
The fact that when the base is io the mantissa of the logarithm is independent of the position of the decimal point in the number affords the chief reason for the choice of io as base.
In tables of logarithms of numbers to base io the mantissa only is in general tabulated, as the characteristic of the logarithm of a number can always be written down at sight, the rule being that, if the number is greater than unity, the characteristic is less by unity than the number of digits in the integral portion of it, and that if the number is less than unity the characteristic is negative, and is greater by unity than the number of ciphers between the decimal point and the first significant figure.
The next great advance on the Trigonometria artificialis took place more than a century and a half afterwards, when Michael Taylor published in 1792 his seven-decimal table of log sines and tangents to every second of the quadrant; it was calculated by interpolation from the Trigonometria to 10 places and then contracted to 7.
These extracts contain all the original statements made by Napier, Robert Napier and Briggs which have reference to the origin of decimal logarithms. It will be seen that they are all in perfect agreement.
Napier thus had complete command over decimal fractions and the use of the decimal point.
In 1782 he prepared an elaborate report on the coinage, suggesting the use of the decimal system and of the terms dollar and cent.
The Egyptians report the weight of a measure of various articles, amongst others water (6), but lay no special stress on it; and the fact that there is no measure of water equal to a direct decimal multiple of the weight-unit, except very high in the scale, does not seem as if the volume was directly based upon weight.
This is also evidently the Olympic cubit; and, in pursuance of the decimal multiple of the digit found in Egypt and Persia, the cubit of 25 digits was (1/4)th of the orguia of 100 digits, the series being --
-- The ounce (480 gr.) and multiples and decimal parts of the ounce troy from 500 ounces to 0.001 oz.
When the base is to, the logarithms of all numbers in which the digits are the same, no matter where the decimal point may be, have the same mantissa; thus, for example, log 2.5613 =0-4084604, log 25.613 =1.4084604, log 2561300 = 6.4084604, &c.
Briggs appreciated clearly the advantages of a centesimal division of the quadrant, and by dividing the degree into hundredth parts instead of into minutes, made a step towards a reformation in this respect, and but for the appearance of Vlacq's work the decimal division of the degree might have become recognized.
Passing now to the invention of common or decimal logarithms, that is, to the transition from the logarithms originally invented by Napier to logarithms to the base io, the first allusion to a change of system occurs in the "Admonitio " on the last page of the Descriptio (1614), the concluding paragraph of which is " Verum si huius inventi usum eruditis gratum fore intellexero, dabo fortasse brevi (Deo aspirante) rationem ac methodum aut hunc canonem emendandi, aut emendatiorem de novo condendi, ut ita plurium Logistarum diligentia,limatior tandem et accuratior, quam unius opera fieri potuit, in lucem prodeat.
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.
The coinage of Mexico, now concentrated at the mint in the capital (all others having been closed) is based (since November 28, 1867) on the decimal system - the peso being divided into 100 centavos - and consists of gold, silver, nickel and bronze coins, whose weight and fineness are determined by the monetary law of 1904.
Since 1871 the decimal system of coinage, corresponding to that of the United States, has been the only one employed.
The Sum Will Be The Year Of The Christian Era, And The Day Of The Year Will Be Found By Multiplying The Decimal Figures By 365.
The decimal point is, however, used systematically in the Constructio (1619), there being perhaps two hundred decimal points altogether in the book.
He again sat on the commission of 1799 for the construction of the metric system, and by his zealous advocacy of the decimal principle largely contributed to its adoption.
To him are due the introduction of the decimal system of currency and the adoption of a system of protection to Canadian manufactures.
The banking system of the country was put on a sound footing by a series of acts culminating in 1871, and in the same year a uniform system of decimal currency was established for the whole Dominion.
It was a long time before decimal arithmetic came into general use, and all through the 17th century exponential marks were in common use.
There seems but little doubt that Napier was the first to make use of a decimal separator, and it is curious that the separator which he used, the point, should be that which has been ultimately adopted, and after a long period of partial disuse.
The voters were to choose one-tenth of their number (notabilities of the commune); one-tenth of these would form the notabilities of the department; while by a similar decimal sifting, the notabilities of the nation were selected.
He was one of the first members, and became president of the Bureau of Longitudes, took a prominent place at the Institute (founded in 1796), professed analysis at the Ecole Normale, and aided in the organization of the decimal system.
R= io, we get the ordinary expression of P/Q as an integer and a decimal; but, if P/Q were equal to 1/3, we could not express it as a decimal with a finite number of figures.
4d.) is the standard coin, with a subsidiary decimal coinage.
Troy weight was abolished, from the 1st of January 1879, by the Weights and Measures Act 1878, with the exception only of the Troy ounce, its decimal parts and multiples, legalized in 1853, 16 Vict.
(arithmetic) elementary lessons on the notation of decimal fractions.
Decimal fractions had been employed for the extraction of square roots.
Briggs's Logarithmorum chilias prima, which contains the first published table of decimal or common logarithms, is only a small octavo tract of sixteen pages, and gives the logarithms of numbers from unity to 1000 to 14 places of decimals.
Decimal or Briggian Antilogarithms. - In the ordinary tables of logarithms the natural numbers are all integers, while the logarithms tabulated are incommensurable.
The two systems of logarithms for which extensive tables have been calculated are the Napierian, or hyperbolic, or natural system, of which the base is e, and the Briggian, or decimal, or common system, of which the base is io; and we see that the logarithms in the latter system may be deduced from those in the former by multiplication by the constant multiplier /loge io, which is called the modulus of the common system of logarithms.