of the second often not much, are his who has to provide for his daily bread by his daily labour. Hence some of the greatest names, in all departments of philosophy and learning, are those of persons who, unembarrassed by the toils and cares of obtaining a subsistence, have been free to lead a life of contemplation, having purchased to themselves that inestimable privilege by a relinquishment of the other pleasures or objects of ambition, ordinarily followed by those in their situation, and seeking no other advantage from their riches or their competence than that of being able to devote their time and their powers of mind to labours of their own choosing.

The first name we shall mention is that of JOHN NAPIER, often, but erroneously, called Lord Napier. He was not a nobleman, but only what would in England be called a lord of a manor. In 1562 he entered St. Salvator's College, St. Andrew's, as appears by the books of the university. At this time he was only twelve years old; but this was not an unusually early age in those times for going to the university in Scotland. Many entered even younger; and in the university of Glasgow it was found necessary to make a law, that no student should be admitted to the degree of bachelor of arts before the age of fifteen, unless upon good reason appearing to dispense with a year in any particular case.

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 long 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 upona political career with many advantages, he declined interfering in public affairs, and retired to Meichis

ton, his family seat, 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 shown. 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, foretelling 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."

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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 original 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 greal 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 had refuted its conclusions.

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 en tered 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 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 negotiations 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; 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 plain 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 everybody 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 either one of two things; 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