upon good reason appearing to dispense with a year in any particular case. Napier's own language, therefore, is not inappropriate, when, in his treatise on the Apocalypse, published in 1593, he tells us that it is "no few years" since he began to " precogitate the same," being then, as he adds, in his "tender years and barneage (childhood) at St. Androes, at the schools.'

On leaving college, Napier is understood to have set out on his travels, in the course of which he visited France, Italy, and Germany. It is not known when he returned home; but he was probably a considerable time abroad, since we hear nothing farther of him till he was above forty years of age. On arriving again in his own country, although he had already acquired considerable reputation for abilities and learning, and might probably have entered upon a political career with many advantages, he declined interfering in public affairs, and retired to Merchiston, with the intention of devoting himself exclusively to study. A room in which he used to seclude himself for this purpose, at the top of the old tower of Merchiston, is still shewn. He also resided occasionally at Gartness, where he was looked upon by the common people, we are told, as a wizard -a common fate of learned and studious men, down even to an age so recent as this, although Napier's is probably one of the latest names that acquired this species of celebrity. As an evidence that his renown for more than mortal knowledge was not confined to the simple peasantry of Stirlingshire, we may mention that there is preserved in the British Museum, a small tract, printed in London, of which the following is the title: "A Bloody Almanack, fortelling many certaine predictions which shall come to pass this present yeare, 1647; with a calculation

concerning the time of the day of judgment, drawne out and published by that famous astrologer, the Lord Napier of Merchiston."

But the fact is, that although Napier did not himself profess to be either necromancer or astrologer, he cannot be altogether acquitted of pretending to this very insight into futurity which is here attributed to him. The first publication which he gave to the world was an exposition of the Revelations, which appeared at Edinburgh in 1593, prefaced by a dedication to James VI., which is characterized by singular plainness of speech. "Verily and in truth," says the writer, "such is the injury of this our present time, against both the church of God and your majesty's true lieges, that religion is despised and justice utterly neglected; for what by atheists, papists, and cold professors, the religion of God is mocked in all estates; again, for partiality, prolixity, dearth, and deceitfulness of laws, the poor perish, the proud triumph, and justice is nowhere to be found." He then beseeches his majesty to attend himself to these enormities, assuring him that, if he act justly to his subjects, "God will ministrate justice to him against all his enemies, and contrarily, if otherwise." In redressing the evils denounced, he goes on to exhort him to "begin at his own house, family, and court; "a step, the necessity of which he endeavours to impress upon him at considerable length, and with extraordinary intrepidity. There is not a word of flattery in the whole epistle. As for the work itself, it is of a similar character to many others that have been written upon the same mysterious subjects. The most important proposition which it professes to demonstrate is, that the end of the world is to take place some time between the years 1688 and 1700. It is a large and elaborate treatise, and is garnished occasionally with effusions in rhyme, sometimes ori

ginal, and sometimes translated. Among other aids, the author presses the famous Sibylline Oracles into his service, ornamenting them with a metrical version and a commentary. This work appears to have attracted a great deal of attention on its first appearance, and to have retained its popularity for a considerable time. It did not, perhaps, cease to be generally remembered, till the termination of the seventeenth century effectually refuted its conclusions. A fifth edition of it, we observe, appeared at Edinburgh in 1645, which was, perhaps, not the last. It was translated into the French language, and published at Rochelle in 1602*.

Napier's mathematical studies, after all, however, probably did more to procure for him the reputation of being a magician than even these theological lucubrations. It was believed, it seems, that he was attended by a familiar spirit in the shape of a large black dog. A curious anecdote, for the truth of which undoubted evidence exists, would even lead us to suppose that he was not himself averse to being thought in possession of certain powers or arts not shared by ordinary men. A document is still preserved, containing a contract which he entered into, in July 1594, with a brother baron, Logan of Restalrig, to the effect that, "forasmuch as there were old reports and appearances that a sum of money was hid within Logan's house of Fastcastle, John Napier should do his utmost diligence to search and

*Napier's book probably occasioned some controversy. There is a MS. in the British Museum, entitled, "Porta Lucis, or the way to decypher the name, number, and mark of the Beast, by a method more rational, free, and unstrained, than ever any hitherto; occasioned by the peremptore determination of the Lord Napier of Merchistoune, upon the name AaTvos." The only part of the promised treatise, however, which the MS. contains is the Preface, in twelve and a half closely written folio pages.

seek out, and by all craft and ingine to find out the same, and by the grace of God shall either find out the same, or make it sure that no such thing has been there. For his reward he was to have the exact third of all that was found, and to be safely guarded by Logan back to Edinburgh with the same; and in case he should find nothing, after all trial and diligence taken, he refers the satisfaction of his travel and pains to the discretion of Logan*." This, it will be observed, is very cautiously expressed, and so as not distinctly to advance on Napier's part any claim to supernatural skill; but a person engaging in such negociations could hardly be very much surprised, in that age, if he was held to be acquainted with more of the sciences than he chose to admit. The whole affair places before us a very curious picture of the times.

We do not know exactly when it was that Napier deserted theology for mathematics-having in this respect taken just the opposite course to that followed long afterwards by the celebrated Count Swedenborg, who, having been all his previous life a mere man of science, began, when between fifty and sixty years of age, to see visions of the spiritual world, and to converse with angels. But the work upon the Apocalypse was, at any rate, the last of his theological publications. He is understood to have devoted his attention in subsequent years chiefly to astronomy, a science which, recently regenerated by Copernicus and Tycho Brahe, was then every day receiving new illustration from the discoveries of Kepler and Galileo. The demonstrations, problems, and calculations of this science most commonly involve some one or more of the cases of trigonometry, or that branch of the mathematics which, from certain parts, whether sides or angles, of a triangle being given, teaches * Douglas's Peerage, by Wood, ii. 291.

how to find the others which are unknown. On this account trigonometry, both plane and spherical, engaged much of Napier's thoughts; and he spent a great deal of his time in endeavouring to contrive some methods by which the operations in both might be facilitated. Now these operations, the reader, who may be ignorant of mathematics, will observe, always proceed by geometrical ratios, or proportions. Thus, if certain lines be described in or about a triangle, one of these lines will bear the same geometrical proportion to another, as a certain side of the triangle does to a certain other side. Of the four particulars thus arranged three must be known, and then the fourth will be found by multiplying together certain two of those known, and dividing the product by the other. This rule is derived from the very nature of geometrical proportion, but it is not necessary that we should stop to demonstrate here how it is deduced. It will be perceived, however, that it must give occasion, in solving the problems of trigonometry, to a great deal of multiplying and dividing,-operations which, as every body knows, become very tedious whenever the numbers concerned are large; and they are generally so in astronomical calculations. Hence such calculations used to exact immense time and labour, and it became most important to discover, if possible, a way of shortening them. Napier, as we have said, applied himself assiduously to this object; and he was, probably, not the only person of that age whose attention it occupied. He was, however, undoubtedly the first who succeeded in it-which he did most completely by the admirable contrivance which we are now about to explain.

When we say that 1 bears a certain proportion, ratio, or relation to 2, we may mean any one of two things; either that 1 is the half of 2, or that it is less than 2