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

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 Areos.' 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

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 I 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 by 1. If the former be what we mean, we may say that the relation in question is the same as that of 2 to 4, or of 4 to 8; if the latter, we may say that it is the same as that of 2 to 3, or of 3 to 4. Now in

the former case we should be exemplifying what is called a geometrical; in the latter, what is called an arithmetical proportion: the former being that which regards the number of times, or parts of times, the one quantity is contained in the other; the latter regarding only the difference between the two quantities. We have already stated that the property of four quantities arranged in geometrical proportion is, that the product of the second and third, divided by the first, gives the fourth. But when four quantities are in arithmetical proportion, the sum of the second and third, diminished by the subtraction of the first, gives the fourth. Thus, in the geometrical proportion 1 is to 2 as 2 is to 4, if 2 be multiplied by 2 it gives 4; which divided by 1 still remains 4: while in the arithmetical proportion 1 is to 2 as 2 is to 3, if 2 be added to 2 it gives 4; from which if 1 be subtracted, there remains the fourth term 3. It is plain, therefore, that, especially where large numbers are concerned, operations by arithmetical must be much more easily performed than operations by geometrical proportion; for in the one case you have only to add and subtract, while in the other you have to go through the greatly more laborious processes of multiplication and division.

Now it occurred to Napier, reflecting upon this important distinction, that a method of abbreviating the calculation of a geometrical proportion might perhaps be found, by substituting, upon certain fixed principles, for its known terms, others in arithmetical proportion, and then finding, in the quantity which should result from the addition and subtraction of these last, an indication of that which would have