Theologians From Scotland

"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."

"The biggest improvement in arithmetic during the sixteenth and seventeenth centuries was the invention of logarithms. The basic idea was noted by Stifel. In Arithmetica Integra [1544] he observed that the terms of the geometric progression 1, r, r2, r3, ... correspond to the terms in the arithmetic progression 0, 1, 2, 3, ... . Multiplication of two terms in the geometric progression yields a term whose exponent is the sum of the corresponding terms in the arithmetic progression. Division of two terms in the geometric progression yields a term whose exponent is the difference of the corresponding terms in the arithmetic progression. This observation had also been made by Chuquet in Le Triparty en la science des nombres (1484). Stifel extended this connection between the two progressions to negative and fractional exponents. Thus the division of r2 by r3 yields r-1, which corresponds to the term -1 in the arithmetic progression. Stifel, however, did not make use of this connection between the two progressions to introduce logarithms. John Napier, the Scotsman who did develop logarithms about 1594, was guided by this correspondence between the terms of a geometric progression and those of the corresponding arithmetic progression. Napier was interested in facilitating calculations in spherical trigonometry that were being made on behalf of astronomical problems."

"Napier also invented a calculating device known as Napier's bones in 1617 and made common use of decimal multiplications and divisions. The device itself does not use logarithms, but rather is a convenient tool to reduce multiplication and division to a sequence of simple addition and subtraction operations. The method employed by Napier's bones was based on Arab mathematics and Fibonacci's Liber Abaci."

"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."

"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?"

"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."

"It will be admitted... that... artificial helps may prove useful in laborious and protracted multiplications, by sparing the exercise of memory, and preventing the attention from being overstrained. Of this description are the Rods or Bones, which we owe to the early studies of the great Napier, whose life, devoted to the improvement of the science of calculation, was crowned by the invention of logarithms, the noblest conquest ever achieved by man."

"Napier presented his concept of logarithms as a boon to mankind for, as he pointed out, working with logarithms reduces the drudgery of numerical computations enormously."

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

"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."

"SIR, AS the writings of Archimedes were addressed of Sicily, who had perused and relished them, so I do myself the honour, to address to Your Majesty, the following account of the Life, Writings, and Inventions of our British Archimedes..."

"I Have undertaken to write the Life of John Napier, of Merchiston, a man famous all the world over, for his great and fortunate discovery of Logarithms in Trigonometry, by which the ease and expedition in calculation, have so wonderfully assisted the Science of Astronomy, and the arts of practical Geometry and Navigation. Elevated above the age in which he lived, and a benefactor to the world in general, he deserves the epithet of Great. Napier lived in a country of proud Barons, where barbarous hospitality, hunting, the military art, and religious controversy, occupied the time and attention of his contemporaries, and where he had no learned society to assist him in his researches."

"Writing about the middle of the eighteenth century, David Hume proclaimed John Napier of Merchiston as 'the person to whom the title of a great man is more justly due than to any other whom his country ever produced.' This judgment of Hume is the more remarkable, seeing he was himself naturally disposed to exalt literature above science. ...when he awarded the first place among his countrymen to Napier... it was doubtless from an enlivened conviction that his work had been of greater service to humanity."

"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."

"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."

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

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

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

"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."

"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."

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

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

"3. Proposition. The Star and locusts of the fift trumpet, are not the greate Antichrist and his Cleargie, but the Dominator of the Turkes and his armie, who began their dominion in anno Christi 1051."

"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."

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

"Here then (belove reader) thou hast this work devided into two treatises, the first is the said introduction and reasoning, for investigation of the true sense of every cheife Theological tearme and date contained in the Revelation, whereby, not onely is it opened, explained and interpreted, but also that same explanation and interpretation is proved, confirmed and demonstrated, by evidente proofe and coherence of scriptures, agreeable with the event of histories. The seconde is, the principall treatise, in which the whole Apocalyps, Chapter by chapter, Verse by verse, and Sentence by sentence, is both Paraphrastically expounded and Historically applyed. ...And because this whole work of Revelation concerneth most the discoverie of the Antichristian and Papisticall kingdome, I have therefore (for removing of all suspition) in al histories and prophane matters, taken my authorities and cited my places either out of Ethnick auctors, or then papistical writers, whose testimonies by no reason can be refuted against themselves. But in matters of divinitie, doctrine & interpretation of mysteries (leaving all opinions of men) I take me onely to the interpretation and discoverie thereof, by coherence of scripture, and godly reasons following thereupon; which also not only no Papist, but even no Christian may justly refuse. And forasmuch as our scripturs herein are of two fortes, the one our ordinary text, the other extraordinary citations; In our ordinary text, I follow not altogether the vulgar English translation, but the best learned in the Greek tong, so that (for satisfying the Papists) I differ nothing from their vulgar text of S. Jerome, as they cal it, except is such places, where I prove by good reasons, that hee differeth from the Original Greek. In the extraordinary texts of other scriptures cited by me, I followe ever Jeromes latine translation, where any controverse stands betwixt us and the Papists, and that moveth me in divers places to insert his very latine text, for their cause, with the just English thereof, for supply of the unlearned."

"In my tender years and bairn-age, at schools, having on the one part contracted a loving familiaritie with a certain gentleman a papist, and on the other part being attentive to the sermons of that worthy man of God, Maister Christopher Goodman, teaching upon the Apocalyps, I was moved in admiration against the blindness of papists that could not most evidentlie see their seven hilled Citie of Rome, painted out there so lively by Saint John, as the Mother of all Spiritual Whoredome: that not onlie bursted I oute in continuall reasoning against my said familiar, but also from thence forth I determined with myself by the assistance of God's spirit to employ my study and diligence to search out the remanent mysteries of that holy booke (as to this houre praised be the Lord I have bin doing at all such times as convenientlie I might have occasion) &c."

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

"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."

"Therefore, Sir, let it be your Majesty's continuall study (as called and charged thereunto by God) to reforme the universall enormities of your country, and first (taking example of the princely prophet David) to begin at your Majesty's owne house, familie and court, and purge the same of all suspicion of Papists and Atheists and Newtrals, whereof this Revelation foretelleth that the number shall greatly increase in these latter daies. For shall any Prince be able to be one of the destroyers of that great seate, and a purger of the world from Antichristianisme, who purgeth not his owne countrie? Shall he purge his whole countrie who purgeth not his owne house? Or shall he purge his house, who is not purged himself by private meditations with his God?"

"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."

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

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

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

"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."

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

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

"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."

"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."

"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."

"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."

"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."

"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."

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

"The First Proposition. In propheticall dates of daies, weekes, moneths, and yeares, everie common propheticall day is taken for a yeare."

"2. Proposition. The Seven Trumpets of the 8. and 9. chapters, and the Seven Vials of the 16. Chapter, are all one."

"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."

"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."