Their nature. 4. The invention of logarithms is one of the rarest instances of sagacity in the history of mankind; and it has been justly noticed as remarkable, that it issued complete from the mind of its author, and has not received any improvement since his time. It is hardly necessary to say, that logarithms are a series of numbers, arranged in tables parallel to the series of natural numbers, and of such a construction, that by adding the logarithms of two of the latter we obtain the logarithm of their product; by subtracting the logarithm of one number from that of another we obtain that of their quotient. The longest processes therefore of multiplication and division are spared, and reduced to one of mere addition or subtraction.
Property of numbers discovered by Stifelius. 5. It has been supposed that an arithmetical fact, said to be mentioned by Archimedes, and which is certainly pointed out in the work of an early German writer, Michael Stifelius, put Napier in the right course for this invention. It will at least serve to illustrate the principle of logarithms. Stifelius shows that if in a geometrical progression, we add the indices of any terms in the series, we shall obtain the index of the products of those terms. Thus, if we compare the geometrical progression, 1, 2, 4, 8, 16, 32, 64, with the arithmetical one which numbers the powers of the common ratio, namely, 0, 1, 2, 3, 4, 5, 6, we see that by adding two terms of the latter progression, as 2 and 3, to which 4 and 8 correspond in the geometrical series, we obtain 5, to which 32, the product of 4 by 8, corresponds; and the quotient would be obtained in a similar manner. But though this, which becomes self-evident, when algebraical expressions are employed for the terms of a series, seemed at the time rather a curious property of numbers in geometrical progression, it was of little value in facilitating calculation.
Extended to magnitudes. 6. If Napier had simply considered numbers in themselves, as repetitions of unity, which is their only intelligible definition, it does not seem that he could ever have carried this observation upon progressive series any farther. Numerically understood, the terms of a geometrical progression proceed per saltum; and in the series 2, 4, 8, 16, it is as unmeaning to say that 3, 5, 6, 7, 9, in any possible sense, have a place, or can be introduced to any purpose, as that ½, ¼, ⅛, 1/16 or other fractions are true numbers at all.[602] The case, however, is widely different when we use numbers as merely the signs of something capable of continuous increase or decrease of space, of duration, of velocity. These are, for our convenience, divided by arbitrary intervals, to which the numerical unit is made to correspond. But as these intervals are indefinitely divisible, the unit is supposed capable of division into fractional parts, each of them a representation of the ratio which a portion of the interval bears to the whole. And thus also we must see, that as fractions of the unit bear a relation to uniform quantity, so all the integral numbers, which do not enter into the terms of a geometrical progression, correspond to certain portions of variable quantity. If a body falling down an inclined plane acquires a velocity at one point which would carry it through two feet in a second, and at a lower point one which would carry it through four feet in the same time, there must, by the nature of a continually accelerated motion, be some point between these where the velocity might be represented by the number three. Hence, wherever the numbers of a common geometrical series, like 2, 4, 8, 16, represent velocities at certain intervals, the intermediate numbers will represent velocities at intermediate intervals; and thus it may be said that all numbers are terms of a geometrical progression, but one which should always be considered as what it is—a progression of continuous, not discrete quantity, capable of being indicated by number, but not number itself.
[602] Few books of arithmetic, or even algebra, as far as I know, draw the reader’s attention at the outset to this essential distinction between discrete and continuous quantity, which is sure to be overlooked in all their subsequent reasonings. Wallis has done it very well; after stating very clearly that there are no proper numbers but integers he meets the objection, that fractions are called intermediate numbers. Concedo quidem sic responderi posse; concedo etiam numeros quos fractos vocant, sive fractiones, esse quidam uni et nulli quasi intermedios. Sed addo, quod jam transitur εις αλλο γενος. Respondetur enim non de quot, sed de quanto. Pertinet igitur hæc responsio propriè loquendo, non tam ad quantitatem discretam, seu numerum, quam ad continuam; prout hora supponitur esse quid continuum in partes divisibile, quamvis quidem harum partium ad totum ratio numeris exprimatur. Mathesis Universalis, c. 1.
By Napier. 7. It was a necessary consequence, that if all numbers could be treated as terms of a progression, and if their indices could be found like those of an ordinary series, the method of finding products of terms by addition of indices would be universal. The means that Napier adopted for this purpose were surprisingly ingenious; but it would be difficult to make them clear to those who are likely to require it, especially without the use of lines. It may suffice to say that his process was laborious in the highest degree, consisting of the interpolation of 6931472 mean proportionals between 1 and 2, and repeating a similar and still more tedious operation for all prime numbers. The logarithms of other numbers were easily obtained, according to the fundamental principle of the invention, by adding their factors. Logarithms appear to have been so called, because they are the sum of these mean ratios, λογων αριθμος.
Tables of Napier and Briggs. 8. In the original tables of Napier the logarithm of 10 was 3.0225850. In those published afterwards (1618), he changed this for 1.0000000, making of course that of 100, 2.0000000, and so forth. This construction has been followed since; but those of the first method are not wholly neglected; they are called hyperbolical logarithms, from expressing a property of that curve. Napier found a coadjutor well worthy of him in Henry Briggs, professor of geometry at Gresham college. It is uncertain from which of them the change in the form of logarithms proceeded. Briggs, in 1618, published a table of logarithms up to 1,000, calculated by himself. This was followed in 1624 by his greater work, Arithmetica Logarithmica, containing the logarithms of all natural numbers as high as 20,000, and again from 90,000 to 100,000. These are calculated to fourteen places of decimals, thus reducing the error, which strictly speaking, must always exist from the principle of logarithmical construction, to an almost infinitesimal fraction. He had designed to publish a second table, with the logarithms of sines and tangents to the 100th part of a degree. This he left in a considerably advanced state; and it was published by Gellibrand in 1633. Gunter had as early as 1620 given the logarithms of sines and tangents on the sexagesimal scale, as far as seven decimals. Vlacq, a Dutch bookseller, printed in 1628 a translation of Brigg’s Arithmetica Logarithmica, filling up the interval from 20,000 to 90,000 with logarithms calculated to eleven decimals. He published also in 1633 his Trigonometrica Artificialis, the most useful work, perhaps, that had appeared, as it incorporated the labours of Briggs and Gellibrand, but with no great regard to the latter’s fair advantage. Kepler came like a master to the subject; and observing that some foreign mathematicians disliked the theory upon which Napier had explained the nature of logarithms, as not rigidly geometrical, gave one of his own to which they could not object. But it may probably be said that the very novelty to which the disciples of the ancient geometry were averse, the introduction of the notion of velocity into mathematical reasoning, was that which linked the abstract science of quantity with nature, and prepared the way for that expansive theory of infinites which bears at once upon the subtlest truths that can exercise the understanding, and the most evident that can fall under the senses.
Kepler’s new geometry. 9. It was, indeed, at this time that the modern geometry, which, if it deviates something from the clearness and precision of the ancient, has incomparably the advantage over it in its reach of application, took its rise. Kepler was the man that led the way. He published, in 1615, his Nova Stereometria Doliorum, a treatise on the capacity of casks. In this he considers the various solids which may be formed by the revolution of a segment of a conic section round a line which is not its axis, a condition not unfrequent in the form of a cask. Many of the problems which he starts he is unable to solve. But what is most remarkable in this treatise is that he here suggests the bold idea, that a circle may be deemed to be composed of an infinite number of triangles, having their bases in the circumference, and their common apex in the centre; a cone, in like manner, of infinite pyramids, and a cylinder of infinite prisms.[603] The ancients had shown, as is well known, that a polygon inscribed in a circle, and another described about it, may, by continual bisection of their sides, be made to approach nearer to each other than any assignable differences. The circle itself lay, of course, between them. Euclid contents himself with saying that the circle is greater than any polygon that can be inscribed in it, and less than any polygon that can be described about it. The method by which they approximated to the curve space by continual increase or diminution of the rectilineal figure was called exhaustion, and the space itself is properly called by later geometers the limit. As curvilineal and rectilineal spaces cannot possibly be compared by means of superposition, or by showing that their several constituent portions could be made to coincide, it had long been acknowledged impossible by the best geometers to quadrate by a direct process any curve surface. But Archimedes had found, as to the parabola, that there was a rectilineal space, of which he could indirectly demonstrate that it was equal, that is, could not be unequal, to the curve itself.
[603] Fabroni, Vitæ Italorum, i., 272.
Its difference from the ancient. 10. In this state of the general problem, the ancient methods of indefinite approximation having prepared the way, Kepler came to his solution of questions which regarded the capacity of vessels. According to Fabroni, he supposed solids to consist of an infinite number of surfaces, surfaces of an infinity of lines, lines of infinite points.[604] If this be strictly true, he must have left little, in point of invention, for Cavalieri. So long as geometry is employed as a method of logic, an exercise of the understanding on those modifications of quantity which the imagination cannot grasp, such as points, lines, infinites, it must appear almost an offensive absurdity to speak of a circle as a polygon with an infinite number of sides. But when it becomes the handmaid of practical art, or even of physical science, there can be no other objection, than always arises from incongruity and incorrectness of language. It has been found possible to avoid the expressions attributed to Kepler; but they seem to denote in fact nothing more than those of Euclid or Archimedes; that the difference between a magnitude and its limit may be regularly diminished, till, without strictly vanishing, it becomes less than any assignable quantity, and may consequently be disregarded in reasoning upon actual bodies.
[604] Idem quoque solida cogitavit ex infinito numero superflcierum existere, superficies autem ex lineis infinitis, ac lineis ex infinitis punctis. Ostendit ipse quantum ea ratione brevior fieri via possit ad vera quædam captu difficiliora, cum antiquarum demonstrationum circuitus ac methodus inter se comparandi figuras circumscriptas et inscriptas iis planis aut solidis, quæ mensuranda essent, ita declinarentur. Ibid.