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It is remarked by his chief biographer, when Napier and the delegates were ushered into the royal presence: "It must have been a scene worthy of historical painting-this interview between the grotesque King of Scotland and the recluse philosopher. We may imagine the monarch, as portrayed in that ancient description of him. which seems to have been drawn by an actual observer. "Of a middle stature, more corpulent through his clothes than in his body, yet fatt enouch, his clothes ever being made large and easie, the doublets quilted for steletto proofe, his breeches in grate pleits, and full stuffed; of a timorous disposition, which was the greatest reasone of his quilted doublets; his eyes large, ever roulling after aney stranger cam in his presence; in so much as maney, for shame, have left the roome as being out of countenance: his beard werey thin; his toung too large for his mouth, &c.,'-confronted with John Napier, with his serene presence, thoughtful eye, and ample beard, rarely seen within the royal circle.'

On the 29th January, 1593, John Napier wrote an epistle1 to King James against his collusively favouring the Papists, and urged reforms both in Church and State. This epistle he also prefixed as the dedication to that monarch of "The Plain Discovery," which was published the same year. It is described by one of his biographers as containing "remonstrance without sedition, rebuke without disloyalty, and admonition without impertinence." "The Plain Discovery" was translated into French, and three editions were published at Rochelle, the first in 1602.

After the detection of the Spanish plot, Napier was engaged in the invention of some plans and machines for the defence of the island, which were communicated by King James's ambassadors to the English Government. A description of them is preserved in his handwriting, and bearing his signature. The paper is prefaced by the words: "Anno Domini 1596, the 7 June, Secrett Inventionis, profitabill and necessary in theis dayes for defence of this Iland, and withstanding of strangers, enemies of God's truth and religion."

How John Napier was led to the invention of logarithms appears from his own account of the matter. He had made several improvements in trigonometry: two of the most important have since been connected with his name "Napier's Rules," and "Napier's Analogies." He became desirous of finding out some method by which he could abridge the labour of numerical computations connected with this subject. In the year 1614, when he was above sixty years of age, he

1 The following is a short extract from the epistle :-"Praying your Majesty to attend yourself unto these enormities, and (without casting over the credite thereof to wrong wresters of justice), your Majesty's self to wit, certainly that justice be done to these your true and godly lieges, against the enemies of God's church, and their most cruel oppressors. Assuming your Majesty be concordance of al Scriptures, that if your Majesty ministrate justice to them, God the supreme judge shall ministrate justice to you against al your enemies, and contrarily, if otherwise. Therefore, sir, let it be your Majesty's continual study (as called and charged thereunto by God) to reform the universall enormities of your country, and to begin at your Majesty's owne house, family, and court, and purge the same of all suspicion of Papists and Atheists, or 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 secte, and a purger of the world from Antichristianisme, who purgeth not his owne country? shall he purge his whole country who purgeth not his own house? or shall he purge his house, who is not purged himselfe ?"

published his discovery at Edinburgh in a work entitled, "Mirifict Logarithmorum Canonis Descriptio," reserving his method of constructing the tables until he knew what the learned might think of his invention. The work is dedicated to Prince Charles, the son of King James. The dedication opens with the following expressive senti

ment:

"Seeing there is neither study nor any kind of learning that dotl more actuate and stir up generous and heroical wits to excellent and eminent affairs; and contrariwise that doth more deject and keep down sottish and dull minds than the mathematics; it is no marvel that learned and magnanimous princes in all former ages have taken great delight in them, and that unskilful and slothful men have always pursued them with most cruel hatred, as utter enemies to their ignorance and sluggishness." And further adds, "And therefore this invention (I hope) will be much the more acceptable to your Highness, as it yieldeth a more easy and speedy way of accompt. For what car be more delightful and more excellent in any kind of learning than to despatch honourable and profound matters exactly, readily, and without loss of either time or labour?"

The preface of the work is interesting, being addressed to students. of the mathematics. The following copy is taken from the English translation of Napier's work, as it contains some additions made by the author himself:

Seeing there is nothing (right well beloved students in the mathematickes) that is so troublesome to mathematicall practise, nor that doth more molest and hinder calculators, than the multiplications, divisions, square and cubical extractions of great numbers, which, besides the tedious expense of time, are for the most part subject to many slippery errors. I began, therefore, to consider in my minde, by what certaine and ready art I might remove those hindrances. And having thought upon many things to this purpose, I found at length some excellent briefe 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 rootes, doth also cast away from the worke itselfe, even the very numbers themselves that are to be multiplied, divided, and resolved into rootes, and putteth other numbers in their place, which perform as much as they can do, only by addition and subtraction, 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 Latine for the publique use of mathematicians. But now some of our countrymen in this island well affected to these studies, and the more publique good, procured a most learned mathematician to translate the same into our vulgar English tongue, who after he had finished it sent the coppy of it to me, to be seene and considered on by myself.

"I having most willingly and gladly done the same, finde it to be most exact and precisely conformable to my minde and the originall. Therefore it may please you who are inclined to these studies, to receive it from the translator with as much good will as we recommend it unto you. Fare ye well."

This work contains the natural sines and the logarithms of the sines for every minute of the quadrant, with a description and explanation

of the uses of the tables. A translation1 of this work2 was made into English by Edward Wright, M.A., Fellow of Gonville and Caius College, Cambridge, who died in 1615. The translation, however, was published in 1618 by his son Samuel Wright (then a scholar of the same college), and dedicated to the East India Company.

A preface to the reader was added by Henry Briggs, and a table to finde the part proportionall." Henry Briggs was educated at St. John's College, Cambridge; admitted to the degree of M.A. in 1585; elected Fellow in 1588, and appointed reader on Dr. Linacre's foundation in 1592. In 1596 he was chosen the first reader in Geometry in Gresham College, London, and afterwards, in 1619, he was appointed the first Savilian professor of geometry at Oxford. Soon after the publication of the "Canon Mirificus," Briggs communicated in his lectures at Gresham College the improvement of making 1 the logarithm of the ratio of 10 to 1, instead of 2.30258, as Napier had done. And from the evidence that exists, it appears that he was the first person who publicly made known the advantages of this change in the scale, which he also communicated to Napier himself.

In a letter to Mr. (afterwards Archbishop) Usher, of the date of 10th March, 1615, he writes: "Napier, lord of Markinston, 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 which pleased me better and made me more wonder. I purpose to discourse with him concerning eclipses, for what is there we may not hope for at his hands." Accordingly he visited Napier in the summer of 1615. Their first meeting was described to William Lilly, who has thus recorded the account in his "Life and Times."

"I will acquaint you with one memorable story related unto me by John Marr, an excellent mathematician and geometrician, whom I conceive you remember. He was servant to King James I. and Charles I. At first, when the Lord Napier, or Marchiston, made public his logarithms, Mr. Briggs, then reader of the Astronomy Lectures at Gresham College, in London, was so surprised with admiration of them, that he could have no quietness in himself until he had seen that noble person the Lord Marchiston, whose only invention they He acquaints John Marr herewith, who went into Scotland

were.

1 The following "Admonition" does not appear in the original Latin, page 22, sect. 9, cap. 4. It was most probably added by Napier himself when he revised Wright's translation. It refers to that system of the logarithms of the natural numbers in which 1 is made the logarithm of the ratio of 10 to 1. "An Admonition. But because the addition and subtraction of these former numbers may seem somewhat painful, I intend (if it shall please God) in a second edition, to set out such logarithms as shall make those numbers above written to fall upon decimal numbers, such as 100,000,000, 200,000,000, 300,000,000, &c., which are easie to be added or abated to or from any other number."

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2 The complete title-page is:-"A description of the admirable Table of Logarithmes :—with a declaration of the most Plentifull, Easie, and Speedy use thereof in both kinds of Trigonometry, as also in all Mathematical Calculations. Invented and published in Latine by that Honourable Lord John Nepair, Baron of Marchiston, and translated into English by the late learned and famous Mathematician, Edward Wright. With an addition of the Instrumentall Table to finde the part Proportionall, intended by the Translator, and described in the end of the Booke, by Henrie Brigs, Geometry-reader at Gresham House in London. All perused and approved by the Authour, and published since the death of the Translator. Whereunto is added new Rules for the ease of the student. London, printed for Simon Waterson, 1618."

before Mr. Briggs, purposely to be there when these two learned persons should meet. Mr. Briggs appoints a certain day when to meet at Edinburgh; but failing thereof, the Lord Napier was doubtful he would not come. It happened one day, as John Marr and the Lord Napier were speaking of Mr. Briggs, Ah, John (said Marchiston), Mr. Briggs will not now come. At the very instant one knocks at the gate; John Marr hasted down, and it proved Mr. Briggs, to his great contentment. He brings Mr. Briggs up into my lord's chamber, where almost one quarter of an hour was spent, each beholding the other almost with admiration before one word was spoken. At last Mr. Briggs began: 'My lord, I have undertaken this long journey purposely to see your person, and to know by what engine of wit or ingenuity you came first to think of this most excellent help unto astronomy, viz., the logarithms; but, my lord, being by you found out, I wonder nobody else found it out before, when now known it is so easy. 919

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In the year 1617, shortly before his death, Napier published his Rabdologia." In the dedication is the following passage: "The difficulty and prolixity of calculation, the weariness of which is so apt to deter from the study of mathematics, I have always, with what powers and little genius I possess, laboured to eradicate. And with that end in view I published of late years the Canon of Logarithms wrought out by myself a long time ago, which, casting aside the natural numbers, and the more difficult operations performed by them, substituting in their place others affording the same results, by means of easy additions, subtractions, bisections, and trisections. Of which logarithms, indeed, I have now found out another species much superior to the former, and intend, if God shall grant me longer life, and the possession of health, to make known the method of constructing as well as the manner of using them. But the actual computation of this new canon I have left, on account of the infirmity of my bodily health, to those versant in those studies; and especially to that truly most learned man, Henry Briggs, public Professor of Geometry in London, my most beloved friend."

In the following year Briggs paid a second visit to Napier, and after his return to London printed, in 1617, his "Chilias Prima Logarithmorum," but did not publish it till the next year after the death of Napier, which happened on the 3rd April, 1618; for in his preface Briggs writes, "Why these logarithms differ from those set forth by their most illustrious inventor, of ever respectful memory, in his 'Canon Mirificus,' it is to be hoped his posthumous work will shortly make appear."

The posthumous work was published by his son, Robert Napier, in 1619, with the title of "Mirifici Logarithmorum Canonis Constructio." In the preface, speaking of his father, he writes :-"You have then (benevolent reader) the doctrine of the construction of logarithmswhich here he calls artificial numbers, for he had this treatise beside him composed for several years before he invented the word logarithm [λóyor piμós]-most copiously unfolded, and their nature, accidences, and various adaptations to their natural numbers perspicuously demonstrated. I have also thought good to subjoin to the construction itself a certain appendix, concerning the method of forming another and more excellent species of logarithms, to which the inventor alludes in his epistle prefixed to the 'Rabdologia,' and in

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which the logarithm of 1 is 0.1 I have also published some lucubrations upon the new species of logarithms, by that most excellent mathematician, Henry Briggs, public professor in London, who undertook most willingly the very severe labour of calculating this canon, in consequence of the singular affection that existed between him and my father of illustrious memory. Robert Napier in this volume makes no allusion to the claim of Briggs, as his father, in the "Rabdologia," laid claim to that improvement, and stated that he had committed the execution of it to Briggs. From an expression in Briggs's preface to his "Chilias Prima Logarithmorum," it would appear he expected a recognition of his claim in the posthumous work of Napier. But as it had been passed over in silence, Briggs, in the preface to his "Arithmetica Logarithmica," clearly declared the part he had taken, and that he had first suggested the improvement in his lectures.

In the year 1624 Briggs published his great work "Arithmetica Logarithmica," of which a translation in English appeared in 1631. In the address to his readers he gives the following account of the part he took in the improvement of logarithms, and his great labour in the calculation of the improved tables. "Be not surprised that these logarithms are different from those which that illustrious man, the Baron of Marchiston, published in his Canon Mirificus.' For when explaining publicly the doctrine of them to my auditors at Gresham College, in London, I remarked that it would be much more convenient that O should stand for the logarithm of the whole sine [or radius] as in the 'Canon Mirificus,' but that the logarithm of the tenth part of the same whole sine, namely, of 5° 44′ 21′′, should be 10,000,000,000. And concerning this matter I immediately wrote to the author himself; and as soon as the season of the year, and my public teaching would permit, I went to Edinburgh, where being most kindly received by him, I staid a whole month. But when we began to converse about this change in the system, he said that for some time [dudum] he had been sensible of the same thing, and had desired to accomplish it, but however he had published those that he had already prepared, until he could make others more convenient if his duties and feeble health would permit. But

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1 In this appendix he shows how the logarithms of all composite numbers can be found from the logarithms of prime numbers, and thus describes his method. "In order to find the logarithms of all numbers, it is necessary that the logarithms of some two natural numbers be given, or at least assumed, as in the former first construction 0, or cipher, was assumed for the logarithm of the natural number 1, and 10,000,000,000 for the logarithm of the natural number 10. These, therefore, being given, the logarithm of the natural number 5 (which is a prime number) is sought in this manner. Between 10 and 1 is sought the geometric mean, So between 10,000,000,000 and 0 is sought the arithmetic mean, which is 5,000,000,000. Next between 10 and is taken the geometric mean, which is And similarly between 5,000,000,000 and 10 is taken the arithmetic mean, which is 75,000,000,000.” It will be seen by this process that the successive arithmetic means are the logarithms of the corresponding geometric means. But as these geometric means are not the natural prime numbers, if the process be continued it will be found, after twenty-five operations, that the geometric mean (taking seven places of figures) will be very nearly equal to 9, being defective only by the five millionth part of an unit; and the corresponding arithmetic mean may be taken without sensible error as the logarithm of 9. Thus the logarithm of 9 being known, the logarithm of the prime number 3 is also known.

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