Inventors

"It is no exaggeration to say that the invention of logarithms "by shortening the labours doubled the life of the astronomer." Logarithms were invented by John Napier, Baron of Merchiston, in Scotland. It is one of the greatest curiosities of the history of science that Napier constructed logarithms before exponents were used. To be sure, Stifel and Stevin made some attempts to denote powers by indices, but this notation was not generally known,—not even to Harriot, whose algebra appeared long after Napier's death. That logarithms flow naturally from the exponential symbol was not observed until much later. It was Euler who first considered logarithms as being indices of powers. What then was Napier's line of thought?"

"Napier, in 1614, ...employed the idea of the fluxion of a quantity to picture by means of lines the relation between logarithms and numbers."

"David Stuart & John Minto in "Account of the Life of John Napier of Merchiston," The Edinburgh Magazine, or Literary Miscellany (1787) Vol.6"

"Early in the 1660s, a pair of mathematicians from the British Isles, John Napier and Henry Briggs, jointly introduced, perfected, and exploited the "logarithm," a concept having tremendous practical and theoretical significance. Logarithms have the remarkable property of simplifying such otherwise tedious computations as multiplication, division, and the extraction of roots so that no scientist of sound mind would thereafter go about finding \sqrt[7]{234.65} without the benefit of logarithms."

"The invention of logarithms, without which many of the numerical calculations which have constantly to be made would be practically impossible, was due to Napier of Merchiston. The first public announcement of the discovery was made in his Mirifici Logarithmorum Canonis Descriptio, published in 1614, and of which an English translation was issued in the following year; but he had privately communicated a summary of his results to Tycho Brahe as early as 1594. In the work Napier explains the nature of logarithms by a comparison between corresponding terms of an arithmetical and geometrical progression. He illustrates their use, and gives tables of the logarithms of the sines and tangents of all angles of the first quadrant, for differences of every minute, calculated to seven decimal places. His definition of the logarithm of a quantity n was what we should now express by 10^7 \log_{e} \!\left( \frac{10^7}{n} \right). This work... is the first valuable contribution to the progress of mathematics which was made by any British writer."

"The method by which the logarithms were calculated was explained in the Constructio, a posthumous work issued in 1619: it seems to have been very laborious, and depended either on direct involution and evolution, or on the formation of geometrical means. The method by finding the approximated value of a convergent series was introduced by Newton, Cotes, and Euler. Napier had determined to change the base to one which was a power of 10, but died before he could effect it"

"The rapid recognition throughout Europe of the advantages of using logarithms in practical calculations was mainly due to Briggs, who was one of the earliest to recognize the value of Napier's invention. Briggs at once realized that the base to which Napier's logarithms were calculated were inconvenient; he accordingly visited Napier in 1616, and urged the change to a decimal base, which was recognized by Napier as an improvement. On his return Briggs immediately set to work to calculate tables to a decimal base, and in 1617 he brought out a table of logarithms of the numbers from 1 to 1000 calculated to fourteen decimal places."

"The invention of logarithms came on the world as a bolt from the blue. No previous work had led up to it, nothing had foreshadowed it or heralded its arrival. It stands isolated, breaking in upon human thought abruptly without borrowing from the work of other intellects or following known lines of mathematical thought. It reminds me of those islands in the ocean which rise up suddenly from great depths and which stand solitary with deep water close around all their shores. In such cases we may believe that some cataclysm has thrust them up suddenly with earth-rending force. But can it be so with human thought? Did this discovery come as a revelation to Napier, bursting on him as a light from Heaven, or was it the result of slow growth, the evidences of which are now obliterated, like those rocks whose abrupt sides are due, not to sudden and isolated disruption, but to the denudation which has carried away the neighbouring rocks, which, while they remained, testified to the gradual upheaval of the whole?"

"These properties of the indices of numbers were taken notice of by Stifelius, and even by Archimedes in his work on the numbering of the sands; but it is to Baron Napier, of Merchiston, in Scotland, that we are indebted for the happy idea of applying such numbers to the purposes of arithmetical and trigonometrical calculation, which first appeared in his "Mirisici Logarithmorum Canonis Descriptio," published at Edinburgh in 1614. This work was translated by Mr. Edward Wright, and published by his son in 1616. The method of constructing the table was reserved by the ingenious author till the sense of the learned, upon his invention, should be known; nevertheless Kepler, in his "Chilias Logarithmorum," &c. published in 1624; [John] Speidell, in his "New Logarithms," published in 1619; [Benjamin] Ursinius, in his "Table of Logarithms," 1625; and many other mathematicians constructed small tables conformably to the plan of Lord Napier. But of all those who assisted in the construction of logarithmic tables, Briggs is the most conspicuous; it was he who first suggested our present system, the advantages of which are incalculably greater than those first constructed by Napier, at the same time that he laboured more than anyone in the construction of them."

"And if any number of equals to a first sine be multiplied together producing a second, just so many equals to the Logarithm of the first added together produce the Logarithm of the second."

"Any desired geometrical mean between two sines has for its Logarithm the corresponding arithmetical mean between the Logarithms of the sines."

"If a first sine be multiplied into a second producing a third, the Logarithm of the first added to the Logarithm of the second produces the Logarithm of the third. So in division, the Logarithm of the divisor subtracted from the Logarithm of the dividend leaves the Logarithm of the quotient."

"To decrease geometrically is this, that in equal times, first the whole quantity then each of its successive remainders is diminished, always by a like proportional part."

"Whence a geometrically moving point approaching a fixed one has its velocities proportionate to its distances from the fixed one."

"From the Radical table completed in this way, you will find with great exactness the logarithms of all sines between radius and the sine 45 degrees; from the arc of 45 degrees doubled, you will find the logarithm of half radius; having obtained all these, you will find the other logarithms. Arrange all these results as described, and you will produce a Table, certainly the most excellent of all Mathematical tables, and prepared for the most important uses."

"Seeing there is nothing, (right well beloved students of mathematics,) that is so troublesome to mathematical practice, nor that doth more molest and hinder calculations, that the multiplications, divisions, square and cubical extractions of great numbers, which besides the tedious expence of time, are for the most part subject to many slippery errors, I began, therefore, to consider in my mind, by what certain and ready art I might remove these hindrances. And having thought upon many things to this purpose, I found at length some excellent brief rules to be treated of perhaps hereafter: But amongst all, none more profitable than this, which together with the hard and tedious multiplications, divisions, and extractions of roots, doth also cast away even the very numbers themselves that are to be multiplied, divided, and resolved into roots, and putteth other numbers in their place which perform as much as they can do, only by addition and substraction, division by two, or division by three. Which secret invention being, (as all other good things are,) so much the better as it shall be the more common, I thought good heretofore, to set forth in Latin for the public use of mathematicians."

"But now, some of our countrymen in this island, well affected to these studies, and the more public good, procured a most learned mathematician to translate the same into our vulgar English tongue, who after he had finished it, sent a copy of it to me, to be seen and considered on by myself. I having most willingly and gladly done the same, find it to be most exact and precisely conformable to my mind and the original. Therefore it may please you who are inclined to these studies, to receive it from me and the translator, with as much good will as we recommend it unto you.—Fare thee well."

"35 Proposition. The Devils bondage a thousand yeares (cap. 20) is no waies els, but from stirring up of universall warres among nations."

"Conclusion. Then for conclusion, by these interpretative propositions, followeth foure thinges marvelous and notable. First, that the interpretation of every parte of the Revelation, is accessorie or consectarie to other: that is to say, it is so chained and linked together, that every mysterie opens other to the discoverie of the whole. Secondly, that the first halfe of the book is orderly, that is to say, it containeth in order of time the most notable accidents that concerneth Gods Church, from the time of Christs Baptisme successively to the latter day. Thirdly, that every historie prophecied, is limited or dated with his own nŭber of years. Fourthly and last of all, that whatsoever historie is more orderlie and summarlie, than plainly set downe in the first orderlie parte of the booke, the same is repeated, interpreted, or amplified in the last part of the booke: deviding the whole Revelation according to the table following, before we proceed to the principall matter."

"A Logarithmic Table is a small table by the use of which we can obtain a knowledge of all geometrical dimensions and motions in space, by a very easy calculation. It is deservedly called very small, because it does not exceed in size a table of sines; very easy, because by it all multiplications, divisions, and the more difficult extractions of roots are avoided; for by only a very few most easy additions, subtractions, and divisions by two, it measures quite generally all figures and motions."

"34 Proposition. The thousand yeares that Sathan was bound (Revel. 20.) began in Anno Christi 300. or thereabout."

"It is picked out from numbers progressing in continuous proportion. Of continuous progressions, an arithmetical is one which proceeds by equal intervals; a geometrical one which advances by unequal and proportionally increasing or decreasing intervals. Arithmetical progressions: 1, 2, 3, 4, 5, 6, 7, &c.; or 2, 4, 6, 8, 10, 12, 14, 16, &c, Geometrical progressions: 1, 2, 4, 8, 16, 32, 64, &c.; or 243, 81, 27, 9, 3, 1."

"29 Proposition. The name of the beast expressed by the number 666. (cap. 13.) is the name λαγεινος onely."

"30 Proposition. The marke of the Romane beast, is that invisible profession of servitude and obedience, that his subjects hath professed to his Empire, since the first beginning thereof, noted afterward by the Pope, with divers visible markes."

"31 Proposition. The visible marks of the Beast, are the abused characters, of λρς and crosses of all kindes, taken out of the number of the first beasts name."

"28 Proposition. The image of the Beast, is these degenerate Princes, that in name onely were called Roman Emporours, and were neither Romans of blood, nor Emperours of Magnanimitie."

"25 Proposition. The two horned Beast, is the Antichrist and his kingdome, it alone."

"26 Proposition. The Pope is that only Antichrist, prophecied of, in particular."

"27 Proposition. The image, marke, name, and number of the beast: are of the first great Romane beast, and whole Latine impyre universallie, and not of the second beaste, or Antichrist alone in particular."

"32 Proposition. Gog is the Pope, and Magog is the Turkes and Mahometanes."

"33 Proposition. The armies of Gog and Magog (cap. 20) are all one with the armies of the sixt Trumpet and sixt Viall."

"36 Proposition. The 1260 years of the Antichrists universal raign over Christians, begins about the year of Christ 300. or 316. at the farthest."

"So ends this demonstratiue resolution of all difficulties of the Revelation, first of all dates and times, and last of the principall termes and matters, as to the meaner termes and smaller matters, they are interpreted in the notes of the principall treatise."

"24 Proposition. The great ten-horned beast, is the whole bodie of the Latine Empire, whereof the Antichrist is a part."

"16 Proposition. The 42. moneths, 1260 propheticall daies, three great daies and a halfe: And a time, times and halfe a time, signifieth everie one of them, 1260 Julliane yeares."

"17 Proposition. The description of the throne of God in the fourth chapter, is not the description of the majestie of God in heaven, but of his true religion, wherein he is authorised and sits in the throne among his holy elect on earth."

"18 Proposition. The 24. Elders, are the 24 books of the old Testament, and (metonymicè) all the true professors thereof."

"15 Proposition. The 42. moneths, a thousand two hundred and three score propheticall daies, three greate daies and a halfe, and a time, times, and a halfe a time mentioned in Daniel, & in the Revelation, are all one date."

"12 Proposition. The first of the Seven thunders, and the seventh and last Trumpet or Viall, begin both at once in An.1541."

"13 Proposition. Every one of the first three thundering Angels containeth a Jubelee, and then the last foure al at once compleateth the day of judgement."

"19 Proposition. The foure beasts are the foure Evangelles with all the true writers and professors thereof."

"14 Proposition. The day of Gods judgement appears to fall betwixt the yeares of Christ, 1688. and 1700."

"8 Proposition. The first Seal beginneth to be opened in Anno Christi 29. compleat."

"9 Proposition. Everie Seale must containe the Space of Seven yeares."

"10 Proposition. The last Trumpet and Viall beginneth anno Christe 1541 and should end in anno Christi 1786."

"7 Proposition. The last of the Seven Seales, and first of the Seven Trumpets or Vials, begin both at once, in An. 71."

"11 Proposition. The Seven Thunders, whose voices are commanded to bee sealed, and not written (cap.10.4.) are the Seven Angels, specified cap.14. vers. 6.8.9.14.15.17.18."

"5 Proposition. The space of the fift trumpet or vial containeth 245. years, and so much also, every one of the rest of the trumpets or vials doe containe."

"20 Proposition. Gods Temple, although in heaven, is also taken for his holy Church among his heavenly Elect upon the earth, and metonymicè for the whole contents thereof."

"6 Proposition. The first Trumpet or Viall began at the Jubelee, in anno Christi 71."