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little by the experience of history if we do not learn to avoid the errors of that period: and we shall assuredly find the very same principles, so eloquently advocated by Galileo, to be those which alone can effectually secure either religion or science from abuse and perversion.

SECTION IV.

GALILEO. THE

THE CONTEMPORARIES AND SUCCESSORS OF
BACONIAN PHILOSOPHY, AND THE PRECURSORS OF NEWTON.

IN proceeding to trace the advance of science as promoted by the labours of the contemporaries and immediate followers of those great men, whose pre-eminent discoveries have demanded our almost undivided attention in the preceding section, we find various departments of physical research, in their hands assuming a widely extended and highly improved character.

We shall, in the first place, mention some important improvements in mathematics, which belong to the early part of the seventeenth century, and shall then proceed to review the philosophy of Bacon, the promulgation of which constitutes so leading a feature in the literature of the age. We shall afterwards survey the varied labours of that illustrious train of philosophers, who were the disciples of Galileo and Bacon, and the precursors of Newton; including one who added involuntary evidence to the truth of their principles, by the attempt to set up an imposing but delusive theory of a different kind, which shot across the philosophical horizon like a brilliant meteor, containing the elements of its own explosion.

Improvements in Mathematics.

In former periods of scientific history (as we have already had occasion to observe), the progress of pure mathematics had very little connection with that of

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physical knowledge. In the age of which we are now treating, when great changes had begun to operate on the characters of the different branches, and on the relation between them, we find the progress of each much more closely dependent on that of the other. In proportion as the combination of mathematics with physical research was more generally recognised, the advances of the latter created a necessity for some material improvements in the former: and this necessity, once felt, was the most effectual means of calling forth the resources of mathematical genius, to supply those methods which the exigencies of physical investigation demanded.

The discovery of Kepler's laws had exhibited a striking instance of the connection of phenomena by numerical relations; but the discovery had involved an amount of labour, in the mere work of calculation, almost beyond belief. If we look only at the successful theories of the author, the mass of figures through which he worked to establish them was incredible; and these theories only presented some general laws, to which the increasing discoveries of astronomy soon added the necessity of numerous supplementary investigations, all alike requiring toilsome and extensive computations, not only to verify them in the first instance, but afterwards to turn them to practical account in the various applications which were to be founded upon them.

With such quantities, for instance, as the sines and tangents of the tables, taken only to five or six places of decimals, so simple a calculation as merely finding a fourth proportional is extremely troublesome if it occur often; and, for any long series of calculations, occasions a most intolerable sacrifice of labour and, what is more serious, of time; while the extraction of roots and more complex operations, often repeated, would soon become appalling to the most resolute calculator. In fact, with the methods in use about the beginning of the seventeenth century, the single circumstance of the

enormously increasing labour of calculation must almost necessarily have seriously impeded, if not altogether stopped, the progress of astronomical and physical research, which, in other respects, was now beginning to assume so promising an aspect. Thus, some methods of abridging the overwhelming toil of the computer were daily becoming of more imperious necessity; and we now find a remarkable example of the powers of original inventive genius arising to remove these difficulties, and to supply the so much desired instrument of calculation, precisely at the period when it was most needed.

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Whether we choose to estimate the value of inventions by their practical utility, or by their refinement of principle, we must in every sense esteem, as one of the very first importance, the invention of Logarithms, by Napier of Merchiston. He was born of a noble family, in 1550; and enjoyed all the advantages which the best education attainable in those times could bestow. He appears early to have turned his mind to arithmetical and astronomical studies. He soon felt the increasing difficulties we have just mentioned, and was led to try various mechanical devices for abridging these processes. One of these is known by the name of Napier's rods or bones (from the substance of which they were made), which afford an ingenious mechanical help in multiplication. He explained this method in a work called "Rabdologia," in 1616; but, viewing the subject in all its bearings, he soon perceived a far higher mathematical principle, which afforded a method as simple as powerful, and which he not only discussed in theory, but reduced into a practical form.

It has been said, that a hint at least of the principle may be found in the writings of Archimedes: this, however, is only so far true as that a remark of that philosopher exactly serves to suggest the main difficulty which it was Napier's particular merit to overcome.

The first suggestion of the principle may be thus explained: - If we suppose a series of numbers in geometrical progression, any one tolerably conversant with arithmetic will see that, if any two terms in such a progression are multiplied together, the product will also be a term in the same series, and will be found by inspection, if the series has in the first instance been carried far enough. And we can always tell at what term it will be found it will be that term whose number, reckoning from the commencement, is the sum of those of the two terms which are to be multiplied. Thus, to find the product of the third and seventh terms, we have only to take that number which forms the tenth in the series. It is also evident, that these numbers of the terms are also the indices of the powers of the common multiplier, which enter into each term respectively. This is the leading idea which may be supposed suggested by a passage in Archimedes.

Thus, if our computations always involved no other numbers than such as are terms in a geometrical progression, we should only have to add the indices, and thus be led directly to the product; or, conversely, to subtract them, and find the term which is the quotient. Again, by doubling, tripling, &c. the indices, we should find their products indicating a term which would be respectively the square, cube, &c.of the original term; and, conversely, we should have the square, cube, &c. roots. Thus far all was sufficiently clear and simple.

But here arose the main difficulty: this would apply only to a very few limited systems of numbers, and could not be of any general practical utility. The grand discovery of Napier, therefore, amounted to this: that a geometrical progression may be found in which ALL the natural numbers are terms..

Of the methods by which he arrived at this conclusion, or of the general principle on which such a series can be assigned, we cannot here say much; but we may sufficiently illustrate it by an example. If we suppose a geometrical series whose first term is 10 and

whose common multiplier is likewise 10, it is evident that the second term will be 100, or 10 in the power whose index is 2; the third will be 1000, or 10 in the power of 3, &c. But since in algebra the notion of powers includes those whose indices are any numbers, whole or fractional, there may be terms intermediate to these, which shall be powers of 10 whose indices are fractions, or mixed numbers intermediate to the whole indices. Upon this principle we should have, for example, 40, as a term in the series between 10 and 100, which is a power of 10 whose index is the mixed number 1.602. Again, we may carry the series backward to numbers below 10, and we find 5, a power of 10 whose index is the fraction 0.6989; and 1 is, upon a well known algebraical principle, the power of 10 whose index is 0.

In a word, all the natural numbers may find their places, by interpolation somewhere among the terms of such a series, and the corresponding indices are called their LOGARITHMS. * Different systems of logarithms will be formed, according as different geometrical series are assumed. To find the means of assigning such series, and to calculate the indices belonging to the natural numbers, or, conversely, the numbers from the indices, has been the subject of the profound labours of modern analysts, who have devised various methods for the purpose: but the undivided honour of having first accomplished such a work belongs exclusively to Napier.

It has been said, that a hint of some method of the same kind, ascribed to Longomontanus, had been given to Napier; but, from the total absence of all traces of such a principle in the writings of the former, and even of any claim or pretence to it, we can attach no importance to the story. Napier associated in his labours his friend Mr. Briggs, who conversed freely with him on the improvements of which his system was susceptible. Napier, however, published his "Canon * Aoyos, ratio, agilμos, number; the numbers or measures of ratios.

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