and obtain attention to the improvement of its calculating processes and machinery. The railroads will continue to call off large numbers of young men who would otherwise have been employed in counting-houses, and still more if, as we hope will be the case, public attention should discover that, under a sufficient amount of proper direction, there need be no accidents. And here we are reminded that we have got from Napier and Briggs to collisions of railway trains without any abrupt transition; and that there is no saying where we may come at last if we do not return to our subject at once.

The invention of logarithms reduced some of the ordinary operations of arithmetic, in mathematics and its applications, to comparative inutility; in fact, almost placed them on the retired list. This was not a discovery which had to work its way through either neglect or opposition. Ten editions of John Speidell's book of tables, besides pirated reprints, found their way through the press in sixteen years from the time of the announcement, in England alone: independently of what was done by Napier himself, Briggs, and Gunter. Similar encouragement was shown abroad. Throughout every part of the world of calculation except the commercial, the use of logarithms was firmly established in less than thirty years. And it is to be noted that the early calculators were not content with providing only the number of figures which would be held sufficient for ordinary purposes. They presumed that the wants of the arithmetician must be provided for upon a scale of the utmost liberality; and they found him ten places of figures, giving the option, in all questions, of such correctness as is implied in saying that the error need not be more than about one farthing in a million of pounds, or less than a pound on the national debt. Perhaps they overloaded him at first; but Gunter, and those who wrote for ordinary purposes, soon brought the tables down to the seven figures which have ever since been most common. And here it may be remarked, that the practice of putting seven-figure or six-figure tables into the hands of beginners, has always been one of the great stumbling-blocks. The tables are made large dictionaries, and time is wasted in turning over leaves which might be better employed than in conveying the impression that the subject is very repulsive. A card of four-figure logarithms, such as is

now to be easily had, is the best preparation for more extensive tables.

Looking at the progress of logarithms, we have a question to ask which, we will answer for it, has seldom been 'asked before, and never in print: no great warrant, the reader will say, for its wisdom. Never mind; let it be considered on its own merits. How much of the ready reception of Napier's discovery was due to the tables of logarithms, and how much to its giving tables? We will now explain what we mean.

The calculators of the sixteenth century, as we have stated, turned all their attention to trigonometrical tables, the primary importance of which well warranted their devotion. When the logarithms appeared, there were hardly any of those extensive tables of pure arithmetic which have since appeared, and those only recent. Maginus had published the first ten thousand squares in 1592, and Herwart had published the products of numbers up to 1000 times 1000 in 1610, when Napier was actually engaged on his invention. These could have been but little known, and hardly old enough to have had a fair trial. Now, suppose that, at the time of the invention of logarithms, the calculating world had been in full possession and well established use of Herwart's table or Crelle's convenient form of it, of Ludolf's squares up to that of a hundred thousand, of Buchner's cubes up to that of 10,000, of Barlow's fourth and fifth powers up to 1000, and reciprocals up to 10,000, of Chernac's and Buckhardt's divisions, of all the simplifications of trigonometrical theorems and the use of subsidiary angles, and of the methods of the calculus of differences. Each and every one of these things might easily have preceded logarithms. It would then have been difficult to introduce the last-named discovery in one generation: the old stagers would assuredly not have seen any reason to change their methods, and the younger men would soon have learnt that the new discovery must be introduced with discretion, and that large classes of operations would be little the better for it. Now, it is quite the reverse: the tables of logarithms have got such possession of the field that very important tables of other kinds, Barlow's for instance, are little appreciated; and many persons toil at the logarithms to produce results which are ready tabulated for them, or for which at least greater facilitations are provided. A man who was better acquainted with tables

and their history than most of his day, and who was a most accomplished astronomical computer, the late Professor Henderson of Edinburgh, used to say that he found Crelle's multiplication table more useful to him than a table of logarithms. Other kinds of tables have not, in fact, had fair play: the mathematical world has refused every thing except either unassisted calculation or logarithms; to their loss, we believe.

There is only one table which, when wanted at all, is a matter of most absolute necessity, and cannot be replaced by any amount of labour which the most intrepid calculator would think of giving. In our time, even if the trigonometrical tables were all burnt, there is nothing in the calculation of a sine or co-sine which would give more than a morning's work, even though nothing but pen, ink, and paper were supplied. But to determine whether a large number is or is not a prime number, would be the question of a week or a fortnight to the unassisted computer. There are no mathematical laws to aid him, except one, of which it may amuse the novice to hear. Suppose we want to know whether 4764821 has or has not any divisors which divide it without remainder, and that the question is to be solved by a direct and unerring process, without trial. The whole range of mathematical discovery gives but this one test;-Multiply together all the numbers 1, 2, 3...... up to 4764820, one less than the proposed number, add one to the product, and divide by 4764821. Then if there be no remainder to that division, the proposed number has no divisors, or is prime; if there be a remainder, it is not prime. This product would contain somewhere about 30 millions of figures; and of course it would be easier to try every divisor up to 2182, which would be enough, and which, at a minute to each trial, would take 36 hours odd. The existing tables save this trouble for all numbers up to three millions, and it is most fortunate that the question is not one which need often be tried.

The correctness of tables is of course a matter of the utmost importance. The invention of stereotype has much improved us in this respect. In the first place, there is a guarantee that figures shall not be changed while the work is at press. It often happens that some slight disarrangement of type takes place in printing: and the pressmen, instead of calling in a qualified person to replace them, frequently try their own hands, and, as may be expected,

are not always successful. This practice is, of course, forbidden but laws are not effective to prevent every misdemeanour. When the ink was laid on by balls, it was not uncommon for the balls to draw out a type adhering to their moist surfaces. But since the roller has been used for the same purpose, there is less of this danger. Still, except from a stereotyped plate, it is impossible to reckon with the utmost certainty upon the last sheet printed off being a perfect transcript of the first. So well was this known, that, before the Nautical Almanac was stereotyped, it was the practice to read with care the first, the middle, and the last copy of each sheet, after they came from the press, before making up the table of errata. On the utility of stereotyping for preserving the work, without the chance of error arising from new composition, it is unnecessary to speak.

The older tables, though containing many inaccuracies, were nevertheless, all things considered, correct enough. As in many other things, so in numerical printing, a great demand for correctness has arisen in the present century. Of all books, mathematical tables are those in which the printer is of as much consequence as the author. A wrong figure is of as great detriment to the work when it arises in the press, as when it is the calculator's fault. Of many mistakes this cannot be said; for when the error is casual and single, the context may be made to correct it. This is even the case in tables, so far as the leading figures are concerned but the smaller ones (in value) cannot be guessed from those which surround them, without as much trouble as was required for their formation. So that there are these two paradoxes: a bad printer is the same thing as a bad author-and the less consequence a figure is of, the more essential is it that it should be correct.

A little after the beginning of the last century, the disadvantages of moveable types were seen by several publishers of tables; and they adopted a stereotype of their own-namely, the use of copperplate engraving. Several works of this kind exist, in which the whole is from copper. They are principally of the mercantile character; which is the more surprising, when we consider that faith in print has always been the characteristic of the man of business, so far as figures are concerned. A table of interest, or of conversion of one coin into another, advertised by its author as of surpassing correctness, is always taken on his

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word: while no mathematician will receive a new book of the kind without examination.

Calculation is one of the things of which it is uniformly observed that those who are no adepts in it desire that their children should be better off than themselves. The want of it, where it exists, always makes itself felt: while there are many branches of knowledge which are very honestly believed to be useless by those who are without them. Again, the desire to possess the advantage of a coinage which shall facilitate computation in money, that is, a decimal coinage, is, we are well assured, gradually arising among men of business. It has advocates in the public press, and of opposition there is yet none. Το this change we look forward as to one of the greatest benefits which can be conferred, both on education and on the routine of the business of life.

ART. V.-A Catholic History of England. By WILLIAM BERNARD MCCABE. Part I. England; Its Rulers, Clergy, and Poor, before the Reformation, as described by the Monkish Historians. London: Newby, 1847.

F any of our literary friends had consulted us on the selection of a subject on which he might venture to write with any prospect of success, "A History of England" would have been the very last which would have occurred to us. The title had long ceased to convey any definite information to our mind-we looked upon it as composed of cabalistic letters, which might signify a thing filled with paper and print, or with dice and tables-nor, taught by sad experience, do we now ever dare to touch any tome. endorsed with that awful name, lest what appeared to be a book should turn out a backgammon box. We do not blame the toy-shops for their application of the termon the contrary, we think that the selection proves them to be philosophers-and if we might venture an opinion, founded on a tolerably good knowledge of both its applications, we would give it as our decided opinion that there is a great deal more "gammon" in the book than in the box,