Physicists From Scotland

"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?"

"Several years ago (Reader, Lover of the Mathematics) my Father, of memory always to be revered, made public the use of the Wonderful Canon of Logarithms; but, as he himself mentioned on the seventh and on the last pages of the Logarithms, he was decidedly against committing to types the theory and method of its creation, until he had ascertained the opinion and criticism on the Canon of those who are versed in this kind of learning. But, since his departure from this life, it has been made plain to me by unmistakable proofs, that the most skilled in the mathematical sciences consider this new invention of very great importance, and that nothing more agreeable to them could happen, than if the construction of this Wonderful Canon, or at least so much as might suffice to explain it, go forth into the light for the public benefit."

"It may surprise the reader to find this honour claimed for the Inventor of Logarithms, who has hitherto been regarded only on his throne of science, and that by the limited number capable of appreciating his genius. The celebrated historian and philosopher [David Hume] who pronounced him to be the greatest man his country ever produced, founded, probably, none of that estimate upon his theological merits; and more recent authors, ranking high among the historians of Christianity and theological learning in Scotland, have omitted to illustrate their subject with the most efficient example they could have found."

"The undoubted fact that Napier worked for some twenty years at the invention of logarithms before he published his first book relating to them is, to my mind, decisive upon this point. It must have been a slow and gradual evolution, even though that which remains furnishes so few traces of the earlier efforts. Is it then possible, out of what he has left us and out of the circumstances of the times, to read the history of this evolution to reconstitute the process of discovery by deciphering the half-effaced records of its growth?"

"Rather curiously, his works of greatest scientific interest, the Descriptio and Constructio have been most neglected. The former was reprinted in 1620, and also in Scriptores Logarithmici, besides being translated into English. The latter was reprinted in 1620 only. This neglect is no doubt largely accounted for by the advantage for practical purposes of tables computed to the base 10, an advantage which Napier seems to have been aware of even before he had made public his invention in 1614."

"That his invention was the greatest boon genius could bestow upon a Maritime Empire is a truth universally felt, and which no person is better qualified to appreciate than your Majesty. It is a proud reflection for Britain, that she does not owe to a stranger the creation of that intellectual aid which renders your Majesty's Fleets as free and fearless in Navigation as they have ever been in battle."

"From every line of his descent talent seems to have flowed in upon John Napier."

"He was born in the year 1550, at Merchiston, the seat of his forefathers, near Edinburgh; four years after the birth of Tycho, fourteen before Galileo, and twenty one before Kepler."

"Even the sagacity of their author did not see the immense fertility of the principle he had discovered; he calculated his tables merely to facilitate arithmetical, and chiefly trigonometrical computation, and little imagined that he was at the same time constructing a scale whereon to measure the density of the strata of the atmosphere, and the heights of mountains; that he was actually computing the areas and the lengths of innumerable curves, and was preparing for a calculus which was yet to be discovered, many of the most refined and most valuable of its resources. Of Napier, therefore, if of any man, it may safely be pronounced, that his name will never be eclipsed by any one more conspicuous, or his invention superseded by any thing more valuable."

"Napier considered the synchronized motion of two points, each moving on a straight line, the one with constant velocity, and the other with a decreasing velocity proportional to the distance remaining to a fixed point, the initial velocity being the same. In modern notation his model may be written as"

"Among other persons of distinction, who united themselves to him [the earl of Montrose, in support of the royalists and Charles I of England], was Lord Napier of Merchiston, son of the famous inventor of the logarithms, the person to whom the title of GREAT MAN is more justly due, than to any other whom his country ever produced."

"He was distant and isolated from the great arena of letters; cooped up within the narrow limits of desolate Scotland, and encircled with savage sights and sounds of civil discord, above which the name of God was howled by those whose hands were red with murder. When we regard his times, and observe the influence that for so long a period of his life, the war of religion exercised over his intellectual exertions, the wonder is, not that his great contemporaries of the continent became distinguished before him, but that after all he should have extricated his mind from so many toils, and have placed himself by a single effort—though one like the spring of a roused lion—at the side of the astonished demi-gods of science, who had been unconscious of their rival."

"It is probable... that the greatest inventor in science was never able to do more than to accelerate the progress of discovery, and to anticipate what time, "the author of authors," would have gradually brought to light. Though logarithms had not been invented by Napier, they would have been discovered in the progress of the algebraic analysis, when the arithmetic of powers and exponents, both integral and fractional, came to be fully understood. The idea of considering all numbers, as powers of one given number, would then have readily occurred, and the doctrine of series would have greatly facilitated the calculations which it was necessary to undertake. Napier had none of these advantages, and they were all supplied by the resources of his own mind. Indeed, as there never was any invention for which the state of knowledge had less prepared the way, there never was any where more merit fell to the share of the inventor."

"Many computing devices have been used since the invention of the abacus. These include , sector compasses, slide rules, calculators, and computers."

"Let our judgment not be too harsh. The period under consideration is too near the Middle Ages to admit of complete emancipation from mysticism even among scientists. Scholars like Kepler, Napier, Albrecht Duerer, while in the van of progress and planting one foot upon the firm ground of truly scientific inquiry, were still resting with the other foot upon the scholastic ideas of preceding ages."

"Napier, Lord of Merchiston, hath set my head and hands at work with his new and admirable Logarithms. I hope to see him this Summer, if it please God, for I never saw a book [Mirifici Logarithmorum Canonis Descriptio] which pleased me better, and made me more wonder."

"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?"

"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 Church of Scotland was planted by such noblemen as Argyle and Glencairn; such barons as Tullibardine and Grange. It was rendered popular, and thus greatly aided, by such preachers as Knox and Goodman; and it became dignified in the eyes of Protestant Europe by its first and greatest theologian, John Napier."

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

"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."

"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."

"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."

"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."

"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."

"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."

"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."

"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."

"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."

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

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

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

"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."

"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."

"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."

"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."

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

"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."

"22 Proposition. The Woman clad with the Sunne (chap. 12) is the true Church of God."

"23 Proposition. The Whoore, who in the Revelation is Stiled Spirituall Babylon, is not reallie Babylon, but the verie present Citie of Rome."

"21 Proposition. The two witnesses mentioned (Reve.11) are the two Testamĕts, and (metonymicè) the whole true professors thereof."

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

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

"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."

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

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