90 JOHN NAPIER.

whole life in projects and speculations without producing a single article of real
utility, and in other instances hit upon one or two things, perhaps, of the high-
est order of usefulness. As he advanced in. years, he seems to have gradually
forsaken wild and hopeless projects, and applied himself more and more to the
useful sciences. In 1596, he is found suggesting the use of salt in improving
land ; an idea probably passed over in his own time as chimerical, but revived
in the present age with good effect No more is heard of him till, in 16 14, he
astonished the world by the publication of his book of logarithms. He is un-
derstood to have devoted the intermediate time to the study of astronomy, a
science then reviving to a new life, under the auspices of Kepler and Galileo,
the former of whom dedicated his Ephemerides to Napier, considering him as
the greatest man of his age in the particular department to which he applied his
abilities.

" The demonstrations, problems, and calculations of astronomy, most com- monly involve some one or more of the cases of trigonometry, or that branch of mathematics, which, from certain parts, whether sides or angles, of a tri- angle 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, al- ways 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 geometri- cal 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, arid 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 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 produtt 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