LOGARITHMS

History. THE labour and time required for performing the arithmetical operations of multiplication, division, and the extraction of roots, were at one time considerable obstacles to the improvement of various branches of knowledge, and in particular the science of astronomy. But about the end of the sixteenth century and the beginning of the seventeenth, several mathematicians began to consider by what means they might simplify these operations, or substitute for them others more easily performed. Their efforts produced some ingenious contrivances for abridging calculations; but of these the most complete by far was that of John Napier, baron of Merchiston, in Scotland, who invented a system of numbers called logarithms, so adapted to the numbers to be multiplied or divided, that these being arranged in the form of a table, each opposite the number called its logarithm, the product of any two numbers in the table was found opposite that formed by the addition of their logarithms; and, on the contrary, the quotient arising from the division of one number by another was found opposite that formed by the subtraction of the logarithm of the divisor from that of the dividend; and similar simplifications took place in the still more laborious operations of involution and evolution. But before we proceed to relate more particularly the circumstances of this invention, it will be proper to give a general view of the nature of logarithms, and of the circumstances which render them of use in calculation.

Let there be formed two series of numbers, the one constituting a geometrical progression, whose first term is unity or 1, and the common ratio any number whatever; and the other an arithmetical progression, whose first term is 0, and the common difference also any number whatever; for example, suppose the common ratio of the geometrical series to be 2, and the common difference of the arithmetical series 1, and let them be written thus:

Here the terms in the arithmetical series are called the logarithms of the corresponding terms of the geometrical series; that is, 0 is the logarithm of 1, and 1 is the logarithm of 2, and 2 is the logarithm of 4, and 3 that of 8, and so on.

From the manner in which the two series are related to each other, it will readily appear by induction that the logarithms of the terms of the geometrical series have the two following properties:

1. The sum of the logarithms of any two numbers or terms in the geometrical series is the logarithm of that number, or term of the series, which is equal to their product.

For example, let the terms of the geometrical series be 4 and 32; the corresponding terms of the arithmetical series (that is, their logarithms) are 2 and 5: now the product of the numbers is 128, and the sum of their logarithms is 7; and it appears by inspection of the two series, that the latter number is the logarithm of the former, agreeing with the proposition we are illustrating. In like manner, if the numbers or terms of the geometrical series be 16 and 64, the logarithms of which are 4 and 6, we find from the table that $10\; =\; 4\; +\; 6$ is the logarithm of $1024\; =\; 16\; \backslash times\; 64$; and so of any other numbers in the table.

2. The difference of the logarithms of any two numbers or terms of the geometrical series is the logarithm of that term which is the quotient arising from the division of the one number by the other.

Take, for example, the terms 128 and 32, the logarithms of which are 7 and 5; the greater of these numbers divided by the less is 4, and the difference of their logarithms is 2; and by inspecting the two series, this last number will be found to be the logarithm of the former. In like manner, if the terms of the geometrical series be 1024 and 16, the logarithms of which are 10 and 4, we find that $1024\; \backslash div\; 16\; =\; 64$, and that $10\; -\; 4\; =\; 6$; now, in the table, the latter number, viz. 6, is the logarithm of the former, 64.

These two properties of logarithms, the second of which indeed is an immediate consequence of the first, enable us to find with great facility the product or the quotient of any two terms of a geometrical series to which there is adapted an arithmetical series, so that each number has its logarithm opposite to it, as in the preceding short table. For it is evident, that to multiply two numbers, we have only to add their logarithms, and opposite to that logarithm which is the sum we shall find the product required. Thus, to multiply 16 by 128; to 4 the logarithm of 16, add 7 the logarithm of 128, and opposite the sum 11, we find 2048, the product sought. On the other hand, to divide any number in the table by any other, we subtract the logarithm of the divisor from that of the dividend, and look for the remainder among the logarithms, and opposite to it we find the number sought. Thus, to divide 2048 by 128; from 11, the logarithm of 2048, we subtract 7, the logarithm of 128, and opposite the remainder 4, we find 16, the quotient sought.

Let us now suppose any number of geometrical means to be interposed between each two adjoining terms of the preceding geometrical series, and the same number of arithmetical means between every two adjoining terms of the arithmetical series; then, as the results will still be a geometrical and an arithmetical series, the interpolated terms of the latter will be the logarithms of the corresponding terms of the former, and the two new series will have the very same properties as the original series.

If we suppose the number of interpolated means to be very great, it will follow that among the terms of the resulting geometrical series, some one or other will be found nearly equal to any proposed number whatever. Therefore, although the preceding table exhibits the logarithms of 1, 2, 4, 8, 16, &c. but does not contain those of the intermediate numbers, 3, 5, 6, 7, 9, 10, &c. yet it is easy to conceive that a table might be formed by interpolation which should contain, amongst the terms of the geometrical series, all numbers whatever to a certain extent (or at least others very nearly equal to them), together with their logarithms. If such a table were constructed, or at least if such terms

History. of the geometrical progression were found, together with their logarithms, as were either accurately equal to, or coincided nearly with, all numbers within certain limits (for example, between 1 and 100,000), then, as often as we had occasion to multiply or divide any numbers contained in that table, we might evidently obtain the products or quotients by the simple operations of addition and subtraction.

The first invention of logarithms has been attributed by some to Longomontanus, and by others to Juste Byrge, two mathematicians contemporary with Napier; but there is no reason to suppose that either of these anticipated him, for Longomontanus never published any thing on the subject, although he lived thirty-three years after Napier had made known his discovery; and as to Byrge, he is indeed known to have printed a table containing an arithmetical and a geometrical progression written opposite to each other, so as to form in effect a system of logarithms of the same kind as those invented by Napier, without however explaining their nature and use, although it appears from the title he intended to do so, but was probably prevented by some cause unknown to us. But this work was not printed till 1620, six years after Napier had published his discovery, namely, in 1614, in a book entitled Mirifici Logarithmorum Canonis Descriptio; but he reserved the construction of the numbers till the opinion of the learned concerning his invention should be known. It is therefore with good reason that Napier is now universally considered as the first, and most probably the only inventor. His work contains a table of the natural sines and cosines, and their logarithms, for every minute of the quadrant, as also the differences between the logarithmic sines and cosines, which are in effect the logarithmic tangents. There is no table of the logarithms of numbers; but precepts are given, by which they, as well as the logarithmic tangents, may be found from the table of natural and logarithmic sines.

In explaining the nature of logarithms, Napier supposes some determinate line which represents the radius of a circle to be continually diminished, so as to have successively all possible values, and thus to be equal to every sine, one after another, throughout the quadrant. And he supposes this diminution to be effected by a point moving from one extremity towards the other extremity (or rather some point very near it), with a motion that is not uniform, but decreases gradually in such a manner, that if the whole time between the beginning and the end of the motion be conceived to be divided into a very great number of equal portions, the decrements taken away in each of these shall be to one another as the respective remainders of the line. According to this mode of conceiving the line to decrease, it is easy to show that at the end of any successive equal intervals of time from the beginning of the motion, the portions of the line which remain will constitute a decreasing geometrical progression.

Again, he supposes another line to be generated by a point which moves along it equally, or which passes over equal intervals of it in equal times. Thus the portions of the line generated at the end of any equal successive intervals of time from the beginning of the motion will form a series of quantities in arithmetical progression. Now if the two motions be supposed to begin together, the remainders of the one line at the end of any equal intervals of time will form a series of quantities in geometrical progression, while the corresponding portions generated of the other line will constitute a series in arithmetical progression, so that the latter will be the logarithms of the former. And as the terms of the geometrical progression decrease continually from radius, which is the greatest term, to 0, while the terms of the corresponding arithmetical progression increase from 0 upwards, according to Napier's system the logarithm of radius is 0, and the logarithms of the sines

from radius down to 0 are a series of numbers increasing from 0 to infinity.

The velocities or degrees of quickness with which the motions commence may have to each other any ratio whatever, and by assuming different ratios we obtain different systems of logarithms. Napier supposed the initial velocities to be equal; but the system of logarithms produced in consequence of this assumption having been found to have some disadvantages, it has been long superseded by a more convenient one, as we shall presently have occasion to explain.

Napier's work having been written in Latin, was translated into English by Mr Edward Wright, an ingenious mathematician of that period, and inventor of the principles of what is commonly though erroneously called Mercator's sailing. The translation being sent to Napier for his perusal, was returned with his approbation, and with the addition of a few lines, intimating that he intended to make some alterations in the system of logarithms in a second edition. Mr Wright died soon after he received back his translation; but it was published after his death in 1616, accompanied with a dedication by his son to the East India Company, and a preface by Henry Briggs, who afterwards distinguished himself by improving the form or system of logarithms. Mr Briggs likewise gave in this work the description and draught of a scale which had been invented by Wright, as also various methods of his own for finding a logarithm to a given number, and a number to a given logarithm, by means of Napier's table, the use of which had been attended with some inconvenience, on account of its containing only such numbers as were the natural sines to every minute of the quadrant and their logarithms. There was an additional inconvenience in using the table, arising from the logarithms being partly positive and partly negative. The latter of these was, however, well remedied by John Speidell in his New Logarithms, first published in 1619, which contained the sines, cosines, tangents, cotangents, secants, and cosecants, and given in such a form as to be all positive; and the former was still more completely removed by an additional table, which he gave in the sixth impression of his work in 1624, and which contained the logarithms of the integers 1, 2, 3, 4, &c. to 1000, together with their differences and arithmetical complements, &c. This table, which is of great use in finding fluents, is commonly called hyperbolic logarithms, because the numbers serve to express the areas contained between a hyperbola and its asymptote, and limited by ordinates drawn parallel to the other asymptote. This name, however, is certainly improper, as the same spaces may represent the logarithms of any system whatever.

In 1619, Robert Napier, son of the inventor of logarithms, published a second edition of his father's Logarithmorum Canonis Descriptio; and, along with this, the promised Logarithmorum Canonis Constructio, and other pieces written by his father and Mr Briggs. An exact copy of the same two works in one volume was also printed in 1620 at Lyons in France. In 1618 or 1619, Benjamin Ursinus, mathematician to the elector of Brandenburg, published Napier's tables of logarithms in his Cursus Mathematicus, to which he added some tables of proportional parts; and in 1624 he printed his Trigonometria, with a table of natural sines, and their logarithms of the Napierian kind and form, to every ten seconds of the quadrant.

In the same year, 1624, the celebrated John Kepler published, at Marburg, logarithms of nearly the same kind, under the title of Chilias Logarithmorum ad totidem Numeros Rotundos, premissa Demonstratione legitima Ortus Logarithmorum eorumque Usus, &c.; and in the following year he published a supplement to this work. In the

History. preface to this last he says, that several of the professors of mathematics in Upper Germany, and more especially those who were advanced in years, and grown averse to new methods of reasoning which carried them out of their old principles and habits, doubted whether Napier's demonstration of the property of logarithms was perfectly true, and whether the application of them to trigonometrical calculations might not be unsafe, and lead the calculator who should trust in them to erroneous results; and in either case, whether the doctrine were true or not, they scarcely considered Napier's demonstration of it as legitimate and satisfactory. This opinion induced Kepler to compose the above-mentioned work, in which the whole doctrine is treated in a manner strictly geometrical, and free from the considerations of motion to which those elderly Germans had objected.

On the publication of Napier's Logarithms, Mr Henry Briggs, some time professor of geometry in Gresham College, London, and afterwards Savilian professor of geometry at Oxford, applied himself with great earnestness to the study and improvement of them. From the particular view which Napier took of the subject, and the manner in which he conceived logarithms to be generated, it happened that in his system the logarithms of a series of numbers which increased in a decuple ratio (as 1, 10, 100, 1000, &c.) formed a decreasing arithmetical series, whose common difference was 2.3205851. But it occurred to Briggs that it would be better and more conformable to the received decimal notation, to adopt a system in which the logarithms of the terms of such a geometrical series should differ from each other by unity or 1. This idea Briggs communicated to the public in his lectures, and also to Napier himself. He even went twice to Edinburgh to converse with him on the subject; and, on his first visit, Napier said that he had also formerly thought of the same improvement, but that he chose to publish the logarithms he had previously calculated, till such time as his health and convenience would allow him to make others more commodious. And whereas, in the change which Briggs proposed, it was intended to make the logarithms of the sines to increase from 0 (the logarithm of radius) to infinity, whilst the sines themselves should decrease, it was suggested to him by Napier that it would be better to make them increase, so that 0, instead of being the logarithm of radius, should be the logarithm of 1, and that 100,000, &c. should be the logarithm of radius. This Briggs admitted would be an improvement; and having changed the numbers he had already calculated so as to make them suit Napier's modification of his plan, he returned with them next year to Edinburgh, and submitted them to his perusal.

It appears, therefore, that whether Napier or Briggs was the inventor of this improved system of logarithms which has since been universally adopted, Napier had suggested to begin with the low number 1, and to make the logarithms, or the artificial numbers, as he had always called them, to increase with the natural numbers, instead of decreasing; which, however, made no alteration in the figures, but only in their affections or signs, changing them from negative to positive.

On Briggs's return from Edinburgh, in 1617, he printed the first thousand logarithms to eight places of figures, besides the index, with the title of Logarithmorum Chilia prima; but these seem not to have been published till after the death of Napier, for in his preface he expresses a hope that the circumstances which led to a change in the system would be explained in Napier's posthumous work, about to appear. But although Napier had intimated in a note he had given in Wright's translation of the Canon Mirificus, as well as in his Rabdologia, printed in 1617, that he intended to alter the scale, yet he does not state

that Briggs was the first to think of this improvement, or to publish it. And as nothing was said on this point in Napier's posthumous work published in 1619 by his son, Briggs took occasion, in his Arithmetica Logarithmica, to assert his claims to the improvement which he had carried into execution. But he has by no means proved that he himself, and not Napier, was the first who had thought of such improvements.

In 1620, Mr Edmund Gunter published his Canon of Triangles, which contains the artificial or logarithmic sines and tangents to every minute to seven places of figures besides the index, the logarithm of radius being 10. These logarithms are of the kind which had been agreed upon between Napier and Briggs, and they were the first tables of logarithmic sines and tangents that were published of this sort. Gunter also, in 1623, reprinted the same in his book De Sectore et Radio, together with the Chilia prima of Briggs; and in the same year he applied the logarithms of numbers, sines, and tangents, to straight lines drawn upon a ruler. This instrument is now in common use for navigation and other purposes, and is commonly called Gunter's Scale.

The discoveries in logarithms were carried to France by Mr Edmund Wingate, but not first of all, as he says in the preface to his book. He published at Paris in 1624 two small tracts in French upon logarithms, which were reprinted with improvements at London in 1626.

In the year 1624, Briggs published his Arithmetica Logarithmica, a stupendous work, considering the short time he had been in preparing it. He there gives the logarithms of 30,000 natural numbers to fourteen places of figures, besides the index; namely, from 1 to 20,000, and from 90,000 to 100,000, together with the differences of the logarithms. He also gives an ample treatise on their construction and use, and he earnestly solicits others to undertake the computation of the intermediate numbers, offering to give instructions, and paper ready ruled for that purpose, to any person inclined to contribute to the completion of so valuable a work. By this invitation, he had hopes of collecting materials for the logarithms of the intermediate 70,000 numbers, whilst he should employ his time upon the Canon of Logarithmic Sines and Tangents, and so carry on both works at once.

Soon after this, Adrian Vlacy or Flack of Gouda, in Holland, completed the intermediate 70 chiliads, and republished the Arithmetica Logarithmica in 1627 and 1628, with these intermediate numbers, making in all the logarithms of all numbers to 100,000, but only to 10 places of figures. To these was added a table of artificial sines, tangents, and secants, to every minute of the quadrant.

Briggs himself lived also to complete a table of logarithmic sines and tangents, to the 100th part of every degree to fourteen places of figures, besides the index, together with a table of natural sines to the same parts to fifteen places, and the tangents and secants of the same to ten places, with the construction of the whole. But death prevented him from completing the application and uses of them. However, when dying, he committed this to his friend Henry Gellibrand, who accordingly added a preface, and the application of the logarithms to plane and spherical trigonometry. The work was called Trigonometria Britannica, and was printed at Gouda in 1633, under the care of Adrian Vlacy, who in the same year printed his own Trigonometria Artificialis, sive Magnus Canon Trigonometricus Logarithmicus ad Decadas Secundorum Serupulorum Constructus. This contains the logarithmic sines and tangents to 10 places of figures, with their differences for every ten seconds in the quadrant. It also contains Briggs's table of the first 20,000 logarithms to ten places, besides the index, with their differences; and to the whole is prefixed a description of the tables and their applications,

History. chiefly extracted from Briggs's Trigonometria Britannica just mentioned.

Gellibrand also published, in 1635, An Institution Trigonometrical, containing the logarithms of the first 10,000 numbers, with the natural sines, tangents, and secants, and the logarithmic sines and tangents, for degrees and minutes; all to seven decimal places.

The writers whose works we have hitherto noticed were for the most part computers of logarithms. But the system best adapted to practice being now well ascertained, and the labour of constructing the table accomplished, succeeding writers on the subject have had little more to do than to give the tables in the most convenient form. It is true, that in consequence of the numerous discoveries which were afterwards made in mathematics, particularly in the doctrine of series, great improvements were made in the method of computing logarithms; but these, for the most part, came too late to be of use in the actual construction of the tables, although they might be applied with advantage to verify calculations previously performed by methods much more laborious, and to detect various errors which had crept into the numbers.

As it is of importance that such as have occasion to employ logarithms should know what works are esteemed for their extent and accuracy, we shall mention the following:

Sherwin's Mathematical Tables, in 8vo. These contain the logarithms of all numbers to 101,000; and the sines, tangents, secants, and versed sines, both natural and logarithmic, to every minute of the quadrant. The third edition, printed in 1742, which was revised by Gardiner, is esteemed the most correct; but, in the fifth edition, the errors are so numerous, that no dependence can be placed on it.

Gardiner's Tables of Logarithms for all numbers to 101,000, and for the sines and tangents to every ten seconds of the quadrant; also for the sines of the first 72 minutes to every single second, &c. This work, which is in quarto, was printed in 1742, and is held in high estimation for its accuracy. An edition of the same work, with some additions, was printed in 1770, at Avignon, in France; and another by Callet at Paris in 1783, with further improvements. The tables in both are to seven places of figures.

Hutton's Mathematical Tables, containing common, hyperbolic, and logistic logarithms, &c. and much valuable information respecting the history of logarithms, and other branches of mathematics connected with them.

Taylor's Table of Logarithmic Sines and Tangents to every second of the quadrant; to which is prefixed an able introduction by Dr Maskelyne, and a table of logarithms from 1 to 100,000, &c. This is a most valuable work; but being a large quarto volume, and rather expensive, it is less accessible than the preceding, which is an octavo, at a moderate price.

Tables portatives des Logarithmes, contenant les Logarithmes des nombres depuis 1 jusqu'à 108,000; les logarithmes des sinus et tangentes, de seconde en seconde pour les cinq premiers degrés, de dix en dix secondes pour tous les degrés du quart de cercle, et suivant la nouvelle division centésimale de dix-millième en dix millièmes, &c. par Callet. This work is in octavo, and printed in stereotype by Didot.

There are various smaller sets of tables; but probably the most accurate of all are those which Professor Babbage has produced with his very ingenious calculating machine, which has enabled him to detect a variety of errors in former tables. But, what is rather amusing, on examining a set of tables printed in the Chinese character, and which, like every Chinese invention, were older than the deluge, Mr Babbage found they contained

precisely the same errors as those of Vlacq did; thus proving, as had long been suspected, from what source those original inventors had derived their logarithms.

In addition to these, it is proper that we should notice a stupendous work relating to logarithms, originally suggested by the celebrated Carnot, in conjunction with Prieur de la Côte d'Or, and Brunet de Montpelier, about the beginning of the French revolution. This enterprise was committed in 1794 to the care of Baron de Prony, a mathematician of great eminence, who was not only to compose tables which should leave nothing to be desired with respect to accuracy, but to make them the most extended and most striking monument of calculation ever executed or imagined. Two manuscript copies of the work, composed of seventeen volumes large folio, contained, besides an introduction,

1. The natural sines for each 10,000th part of the quadrant, calculated to twenty-five places of decimals, to be published with twenty-two decimals and five columns of differences.

2. The logarithms of these sines, calculated to fourteen decimals, with five columns of differences.

3. The logarithms of the ratios of the sines to the arcs for the first five thousand 100,000th parts of the quadrant, calculated to fourteen decimal places, with three columns of differences.

4. The logarithms of the tangents corresponding with the logarithms of the sines.

5. The logarithms of the ratios of the tangents to the arcs, calculated like those of the third article.

6. Logarithms of numbers from 1 to 100,000, calculated to nineteen places of decimals.

7. The logarithms of numbers from 100,000 to 200,000, calculated to twenty-four decimals, in order to be published to twelve decimals and three columns of differences.

The printing of this work, though begun by the French government, was afterwards suspended.

II.—NATURE OF LOGARITHMS, AND THEIR CONSTRUCTION.

We have already shown that the properties of logarithms are deducible from those of two series, the terms of one of which form a geometrical progression, and those of the other an arithmetical progression; and as this manner of treating the subject is simple, it is perhaps the best adapted of any to such of our readers as have not pursued the study of mathematics to any great extent. We shall now show how, from the same principles, the logarithm of any proposed number may be found, as was done by the earlier computers, though it is far from affording the easiest mode of forming a table.

The first step to be taken in constructing a system of logarithms is to assume the logarithm of some determinate number, besides that of unity or 1, which must necessarily be 0. From the particular view which Napier first took of the subject, he was led to assume unity for the logarithm of the number 2718282, by which it happened that the logarithm of 10 was 2302585; and this assumption being made, the form of the system became determinate, and the logarithm of every number fixed to one particular value.

It was, however, soon observed, that it would be better to assume unity for the logarithm of 10, instead of making it the logarithm of 2718282, as in Napier's first system; and hence the logarithms of the terms of the geometrical progression

1, 10, 100, 1000, 10,000, &c.

were necessarily fixed to the corresponding terms of this arithmetical progression,

0, 1, 2, 3, 4, &c.

That is, the logarithm of 1 being 0, and that of 10 being 1, the logarithm of 100 is 2, that of 1000 is 3, and so on.

The logarithms of the terms of the progression 1, 10, 100, 1000, &c. being thus determined; in order to form the logarithms of the numbers between 1 and 10, and between 10 and 100, and so on, we must conceive a very great number of geometrical means to be interposed between each two adjoining terms of the preceding geometrical series, and as many arithmetical means between the corresponding terms of the arithmetical series; then, like as the terms of the arithmetical series 0, 1, 2, 3, &c. are the logarithms of the corresponding terms of the geometrical series 1, 10, 100, 1000, &c. the interpolated terms of the former will also be the logarithms of the corresponding interpolated terms of the latter. Now, by supposing the number of means interposed between each two terms of the geometrical series to be sufficiently great, some one or other of them may be found which will be very nearly equal to any proposed number. Hence, to find the logarithm of such a number, we have only to seek for one of the interpolated means which is very nearly equal to it, and to take the logarithm of that mean as a near value of the logarithm required.

As a particular example, let it be required to find the logarithm of 5, according to Briggs's system.

First step of the process.—The number 5 is between 1 and 10, the logarithms of which we already know to be 0 and 1: Let a geometrical mean be found between the two former, and an arithmetical mean between the two latter. The geometrical mean will be the square root of the product of the numbers 1 and 10, which is 3.162277; and the arithmetical mean will be half the sum of the logarithms 0 and 1, which is 0.5; therefore the logarithm of 3.162277 is 0.5. But as the mean thus found is not sufficiently near the proposed number, we must proceed with the operation as follows:

Second step.—The number 5, whose logarithm is sought, is between 3.162277, the mean last found, and 10, the logarithms of which we know to be 0.5 and 1; we must now find a geometrical mean between the two former, and an arithmetical mean between the two latter. The one of these is $\backslash sqrt\{(3.162277\; \backslash times\; 10)\}\; =\; 5.623413$, and the other is $\backslash frac\{1\; +\; 0.5\}\{2\}\; =\; 0.75$, the logarithm of 5.623413.

Third step.—We have now obtained two numbers, namely, 3.162277 and 5.623413, one on each side of 5, together with their logarithms 0.5 and 0.75; we therefore, proceeding as before, find the geometrical mean, or $\backslash sqrt\{(3.162277\; \backslash times\; 5.623413)\}$, to be 4.216964, and the arithmetical mean $\backslash frac\{0.5\; +\; 0.75\}\{2\}\; =\; 0.625$, the logarithm of 4.216964.

Fourth step.—We proceed in the same manner with the numbers 4.216964, and 5.623413 (one of which is less and the other greater than 5), and their logarithms 0.625 and 0.75, and find a new geometrical mean, viz. 4.869674, and the corresponding arithmetical mean or logarithm 0.6875.

We must go on in this way till we have found twenty-two geometrical means, and as many corresponding arithmetical means or logarithms. And that we may indicate how these are found from each other, let the numbers 1 and 10 be denoted by A and B, and their geometrical means taken in their order by C, D, E, &c. then the results of the successive operations will be as in the following table:

As the last of these means, viz. Z, agrees with 5, the proposed number, as far at least as the sixth place of decimals, we may safely consider them as very nearly equal, and therefore their logarithms very nearly equal; that is, the logarithm of 5 will be 0.6989700 nearly.

In performing the operations indicated in the preceding table, it is necessary to find the geometrical means at the beginning to many more figures than are here put down, in order to insure at last a result true to 7 decimal places. Thus it appears that the labour of computing logarithms by this method is indeed very great. It is, however, that which was employed by Briggs and Vlacy in the original construction of logarithms; but since their time more easy methods have been found, some of which we shall presently have occasion to explain; and a still easier method by Sir John Leslie will be found at the end of this article.

The logarithm of any number whatever may be found by a series of calculations similar to that just explained. But in constructing the table it would only be necessary to have recourse to this method in calculating the logarithms of prime numbers; for as often as the logarithm of a number which was the product of other numbers, whose logarithms were known, was required, it would be immediately obtained by adding together the logarithms of its factors. On the contrary, if the logarithm of the product of two numbers were known, and also that of one of its factors, the logarithm of the other factor would be obtained from these, by simply taking their difference.

From this last remark it is obvious, that having now found the logarithm of 5, we can immediately find that of 2; for since 2 is the quotient of 10 divided by 5, its logarithm will be the difference of the logarithms of 10 and 5; now the logarithm of 10 is 1, and that of 5 is 0.6989700, therefore the logarithm of 2 is 0.3010300.

Having thus obtained the logarithms of 2 and 5, in addition to those of 10, 100, 1000, &c. we may thence find the logarithms of innumerable other numbers. Thus, because $4\; =\; 2\; \backslash times\; 2$, the logarithm of 4 will be the logarithm of 2 added to itself, or will be twice the logarithm of 2. Again, because $5\; \backslash times\; 10\; =\; 50$, the logarithm of 50 will be the sum of the logarithms of 5 and 10. In this manner it is evident we may find the logarithms of $8\; =\; 2\; \backslash times\; 4$, of $16\; =\; 2\; \backslash times\; 8$, of $25\; =\; 5\; \backslash times\; 5$, and of as many more such numbers as we please.

Besides the view we have hitherto taken of the theory of logarithms, there are others under which it has been presented by different authors. Some of these we proceed to explain, beginning with that in which they are defined to be the measures of ratios; but to see the propriety of this definition, it must be understood what is meant by the measure of a ratio.

According to the usual definition of a compound ratio, if there be any number of magnitudes $A,\; B,\; C,\; D$, in continued proportion, the ratio of the first, $A$, to the third, $C$, is considered as made up of two ratios, each equal the ratio of the first, $A$, to the second, $B$. And in like manner the ratio of the first, $A$, to the fourth, $D$, is considered as made of three ratios, each equal the same ratio of the first to the second, and so on. Thus, to take a particular example in numbers, because the ratio of 81 to 3 may be considered as made up of the ratio of 81 to 27, and of 27 to 9, and of 9 to 3, which three ratios are equal among themselves, the ratio of 81 to 3 will be triple that of 9 to 3; and in like manner the ratio of 27 to 3 will be double that of 9 to 3. Also, because the ratios of 1000 to 100, 100 to 10, 10 to 1, are all equal, the ratio of 1000 to 1 will be three times as great as that of 10 to 1; and the ratio of 100 to 1 will be twice as great; and so on.

Taking this view of ratios, and considering them as a particular species of quantities, made up of others of the same kind, they may evidently be compared with each other in the same manner as we compare lines or quantities of any kind whatever. And as, when estimating the relative magnitude of two quantities, two lines, for example, if the one contains five such equal parts as the other contains seven, we say the one line has to the other the proportion of 5 to 7; so, in like manner, if two ratios be such, that the one can be resolved into five equal ratios, and the other into seven of the same ratios, we may conclude that the magnitude of the one ratio is to that of the other as the number 5 to the number 7; and a similar conclusion may be drawn, when the ratios to be compared are any multiples whatever of some other ratio.

Since lines and other quantities, which admit of no common measure, are said to be incommensurable to each other, the same will obviously happen to ratios; that is, there may be two ratios such, that into whatever number of equal ratios the one is divided, the other cannot possibly be exactly equal to a ratio composed of any integral number of these. We may, however, conceive the number of equal ratios into which the one is divided to be so great, that a certain number of them shall compose a ratio more nearly equal to the other ratio than by any assignable difference. Therefore, like as we can always find numbers which shall have among themselves, either accurately, or as nearly as we please, the same ratios as lines or other magnitudes have to each other, and which therefore may be taken as the measures or representatives of the lines; so also, corresponding to any system of ratios, there may be always found a series of numbers which will have the same proportions among themselves as the ratios have to each other, and which may in like manner be called the measures of the ratios.

Let us now suppose that unity, or 1, is assumed as the common consequent or second term of all ratios whatever; and that the ratio of 10 (or some particular number) to 1 is compounded of a very great number of equal ratios, as, for example, 1,000,000; then, as each of these will be very near the ratio of equality (for it will be the ratio of the first term to the second of a series consisting of one million and one continued proportionals, the first of which is 10 and last 1), it will follow, and is easy to conceive, that the ratios of all other numbers to unity will each be very nearly equal to some multiple of that small ratio. And by supposing the number of small equal ratios of which the ratio of 10 to 1 is composed, to be sufficiently great, the ratios of all other numbers to unity may be as nearly equal to ratios which are multiples of that small ratio, as we please. Let us still suppose, however, for the sake of illustration, that the number of small ratios contained in that of 10 to 1 is 1,000,000; then, as it may be proved that the ratio of 2 to 1 will be very nearly the same as a ratio

composed of 301,030 of these, and that the ratio of 3 to 1 will be nearly equal to a ratio composed of 477,121 of them; and that the ratio of 4 to 1 will be nearly equal to a ratio composed of 602,060 of them, and so on; these numbers, viz. 1,000,000, 301,030, 477,121, and 602,060, or any other numbers proportional to them, will be the measures of the ratios of 10 to 1, 2 to 1, 3 to 1, and 4 to 1, respectively; and the same quantities will also be what have been called the logarithms of the ratios; for the word logarithm, if regard be had to its etymology, is λογαρίθμος, or the numbers of small and equal ratios (or ratiunculae, as they have been called) contained in the several ratios of quantities one to another.

We have, for the sake of illustration, assumed 1,000,000 as the measure of the ratio of 10 to 1, by which it happens, as already observed, that the measures of the ratios of 2 to 1, 3 to 1, &c. are 301,030 and 477,121 respectively; as, however, these measures are not absolute, but relative quantities, we may assume any other numbers whatever instead of these, provided they are proportional to them. Accordingly, we may assume 1 as the measure or logarithm of the ratio of 10 to 1; and then the logarithms of the ratios of 2 to 1, 3 to 1, &c. instead of being 301,030, 477,121, &c. will be .301030 and .477121, &c. respectively, that is, each will be one millionth of what it was before.

In Briggs's system, the logarithm of the ratio of 10 to 1, or, to speak briefly, the logarithm of 10, is unity; but we are at liberty to assume any number whatever, as that whose logarithm shall be unity. Napier, in consequence of his particular views, chose 2,718,282; and hence it happens that the logarithms of the ratios are expressed by different numbers in the two systems.

But, to show the identity of the properties of logarithms, as explained in the two different views now given of the subject, let $A$ and $B$ denote any two numbers. The ratio of their product to unity, that is, the ratio of $A\; \backslash times\; B$ to 1, is compounded of the ratio of $A$ to 1 and of $B$ to 1; and consequently the logarithm of the ratio of $A\; \backslash times\; B$ to 1 will equal the sum of the logarithms of the ratios of $A$ to 1, and of $B$ to 1; or, in other words, the logarithm of $A\; \backslash times\; B$ will be the sum of the logarithms of $A$ and $B$. Now, $\backslash log.\; (A\; \backslash times\; B)\; =\; \backslash log.\; A\; +\; \backslash log.\; B$, therefore, $\backslash log.\; B\; =\; \backslash log.\; (A\; \backslash times\; B)\; -\; \backslash log.\; A$. Let $\backslash frac\{C\}\{D\}$ be substituted for $B$, and $D$ for $A$, then (because $A\; \backslash times\; B\; =\; D\; \backslash times\; \backslash frac\{C\}\{D\}\; =\; C$) we have $\backslash log.\; \backslash frac\{C\}\{D\}\; =\; \backslash log.\; C\; -\; \backslash log.\; D$.

Such is a short sketch of the theory of logarithms as deducible from the doctrine of ratios. It was in this way that the celebrated Kepler treated the subject; and he has been followed by Mercator, Halley, and Cotes, as well as by mathematicians of later times, as by Baron Maseres in his Trigonometry. The same mode was likewise adopted in the posthumous works of Dr Robert Simson. As, however, the doctrine of ratios is very abstract, and the mode of reasoning upon which it has been established is of a peculiar and subtle kind, we presume that the greater number of readers will think this view of the subject less simple and natural than the following, in which we mean to deduce the theory of logarithms, as well as the manner of computing them, from the properties of the exponents of powers.

The common scale of notation in arithmetic is so contrived as to express all numbers whatever by the powers of 10, which is the root of the scale, and the nine digits serving as co-efficients to these powers. Thus, if $R$ denote 10, the root of the scale, so that $R^2$ will denote 100, and $R^3$ 1000, and so on, the number 471,509 is otherwise expressed by $4R^5\; +\; 7R^4\; +\; 1R^3\; +\; 5R^2\; +\; 0R^1\; +\; 9R^0$, which is

equivalent to $4R^5\; +\; 7R^4\; +\; R^3\; +\; 5R^2\; +\; 9$. Again, the mixed number 371.243 is expressed by $3R^5\; +\; 7R^4\; +\; R^3\; +\; \backslash frac\{2\}\{R\}\; +\; \backslash frac\{4\}\{R^2\}\; +\; \backslash frac\{3\}\{R^3\}$, or by $3R^5\; +\; 7R^4\; +\; R^3\; +\; 2R^\{-1\}\; +\; 4R^\{-2\}\; +\; 3R^\{-3\}$. As to vulgar fractions, by transforming them to decimals, they may be expressed in the same manner. Thus $\backslash frac\{3\}\{8\}\; =\; .375\; =\; 3R^\{-1\}\; +\; 7R^\{-2\}\; +\; 5R^\{-3\}$. Also $\backslash frac\{2\}\{3\}\; =\; .666$, &c. $=\; 6R^\{-1\}\; +\; 6R^\{-2\}\; +\; 6R^\{-3\}\; +\; \backslash dots$

But instead of expressing all numbers by the sums of certain multiples of the successive powers of some particular number, we may also express them, if not accurately, at least as near as we please, by a single power, whole or fractional, of any positive number whatever, which may be either whole or fractional, but must not be unity.

Let us take, for example, 2 as the number, by the powers of which all others are to be expressed. Then it may be shown that the numbers 1, 2, 3, &c. are all expressible by the powers of 2, as follows.

and so on. And if instead of 2 we take the number 10, then we have

Hence we may conclude, that if $r$ be put for some determinate number, and $n$ for any indefinite positive number, whole or fractional, it is always possible to find another number $N$, such that $r$ being raised to the power $N$, shall either be exactly equal to $n$, or as near it as we please; that is, we shall have $r^N\; =\; n$.

When numbers are expressed in this way by the powers of some given number $r$, the exponent of that power of $r$ which is equal to any assigned number is called the logarithm of that number. Thus, if $r^N\; =\; n$ ($n$ being put for any number), $N$ is the logarithm of $n$.

The logarithms which are produced by giving to $r$ some determinate value, constitute a system of logarithms; and the constant number $r$, from which the system is formed, is called the base, radix, or radical number of the system.

The properties of logarithms may readily be deduced from the above definition as follows: Let $a$ and $b$ be any two numbers, and $A$ and $B$ their logarithms; then $r$ being supposed to denote the base or radical number of the system, we have $a\; =\; r^A$ and $b\; =\; r^B$; now if we take the product of $a$ and $b$, we have $ab\; =\; r^A\; \backslash times\; r^B\; =\; r^\{A+B\}$; but, according to the definition, $A+B$ is the logarithm of $ab$ (for it is the index of that power of $r$ which is equal to $ab$), therefore the sum of the logarithms of any two numbers $a$ and $b$ is the logarithm of their product $ab$. Again, we

have $\backslash frac\{a\}\{b\}\; =\; \backslash frac\{r^A\}\{r^B\}\; =\; r^\{A-B\}$, but here $A-B$, being the index

of that power of $r$ which is equal to $\backslash frac\{a\}\{b\}$, is the logarithm of $\backslash frac\{a\}\{b\}$;

hence, if one number $a$ be divided by another $b$, the difference between their logarithms is the logarithm of the quotient $\backslash frac\{a\}\{b\}$.

Let $n$ express any number whatever, then raising both sides of the equation $a\; =\; r^A$ to the $n$th power, we have $a^n\; =\; (r^A)^n\; =\; r^\{nA\}$; but here $nA$ is manifestly the logarithm of $a^n$; therefore, the logarithm of $a^n$, any power of a number, is the product of the logarithm of the number by $n$, the index of the power. This must evidently be true, whether $n$ be a whole number or a fraction, positive or negative.

From these properties, it is easy to see in what manner a table exhibiting the logarithms of all numbers within certain limits may be applied to simplify calculations; for since the sum of the logarithms of any two numbers is the logarithm of their product, it follows, that as often as we have occasion to find the product of two or more numbers, we have only to add their logarithms taken from the table into one sum, and to look for the number whose logarithm is equal to that sum, and this number will be the product required. Also, because the difference between the logarithm of the dividend and that of the divisor is the logarithm of the quotient, whenever we have occasion to divide one number by another, we have only to subtract the logarithm of the divisor from that of the dividend, and opposite to that logarithm in the table, which is the remainder, we find the quotient.

As the logarithm of any power of a number is the product of the logarithm of the number by the index of the power; and, on the contrary, the logarithm of any root of a number is the quotient found by dividing the logarithm of the number by the index of the root; it follows that we may find any power or root of a number by multiplying the logarithm of the number by the index of the power, or dividing it by the index of the root, and taking that number in the table whose logarithm is the product or quotient for the power or root required.

If in the equation $a\; =\; r^A$ (where $a$ is any number, $A$ its logarithm, and $r$ the base of the system) we suppose $a=1$, then $r^A=1$; but this equation can only be satisfied by $A=0$. Hence it appears, that in every system of logarithms, the logarithm of unity must be 0. If, on the other hand, we assume $a=r$, then we have the equation $r=r^A$, which is immediately satisfied by putting $A=1$; therefore, the logarithm of the base, or radical number of every system, is necessarily unity.

If we suppose $r$ and $a$ to be each a positive number greater than unity, then $A$ will be a positive number; for

if it be a negative we would have $a\; =\; r^A\; =\; \backslash frac\{1\}\{r^\{|A|\}\}$, a pro-

per fraction, and at same time a number greater by hypothesis than unity, which is absurd. If, on the contrary, we suppose $a$ a proper fraction, then $A$ must necessarily be negative, otherwise $r^A$ would be greater than unity, and $a\; (=\; r^A)$ also greater than unity, while by hypothesis it is a fraction less than unity, which is absurd. Therefore, in every system, the base of which exceeds unity, the logarithm of a whole or mixed number is always positive, but the logarithm of a proper fraction is always negative.

Because the logarithm of $r$ is unity, that of $r^n$ will be $n$; therefore, the logarithm of any integral power of the radical number $r$ will always be an integer.

Let $r$ and $r\text{'}$ denote bases of two different systems; and let $A$ be the logarithm of a number, $a$, taken according to the first of these, and $A\text{'}$ its logarithm according to the second. Then, because $a\; =\; r^A$, and $a\; =\; r\text{'}^\{A\text{'}\}$, it follows that $r^A\; =\; r\text{'}^\{A\text{'}\}$, and $r\; =\; r\text{'}^\{\backslash frac\{A\text{'}\}\{A\}\}$. Let us now suppose that $r\text{'}\text{'}$ is the base of a third system of logarithms, and $R$ and $R\text{'}$,

Nature of Logarithms. the logarithms of $r$ and $r\text{'}$, taken according to this third system; then, because

therefore $r^R\; =\; r^\{R\text{'}\}$, and $r\; =\; r^\{\backslash frac\{R\text{'}\}\{R\}\}$; but we have already found $r\; =\; r^\{\backslash frac\{A\text{'}\}\{A\}\}$, therefore $r^\{\backslash frac\{A\text{'}\}\{A\}\}\; =\; r^\{\backslash frac\{R\text{'}\}\{R\}\}$, and consequently

Hence it appears, that the logarithm of a number, taken according to one system, has to its logarithm, taken according to any other system, a constant ratio, which is the same as that of the reciprocals of the logarithms of the radical numbers of those systems.

Let us next suppose that $a$ and $b$ are two numbers, and $A$ and $B$ their logarithms, taken according to the same system, and $r$ the base of the system; then, because

therefore $a^B\; =\; b^A$, and $a\; =\; b^\{\backslash frac\{A\}\{B\}\}$. Now as $r$ is not found in this equation, the value of the fraction $\backslash frac\{A\}\{B\}$ depends only on $a$ and $b$; therefore, the logarithms of any two given numbers have the same ratio to each other in every system.

Having now explained the properties which belong to the logarithms of any system, we proceed to investigate general rules by which the number corresponding to any logarithm, and, on the contrary, the logarithm corresponding to any number, may be found the one from the other. And for this end let us denote any number whatever by $y$, and its logarithm by $x$, and put $r$ as before for the base or radical number of the system; then, by the nature of logarithms,

Put $r\; =\; 1\; +\; a$, and let the expression $(1\; +\; a)^x$ be expanded into a series by the binomial theorem; thus

Let this series, the terms of which are arranged according to the powers of $a$, be transformed into another, whose terms shall be arranged according to the powers of $x$; and to effect this we must find the actual products of the factors which constitute the powers of $a$, and arrange the terms anew, as follows:

so that adding into one sum the quantities on each side of the sign $=$, and recollecting that the sum of those on the left-hand side is equal to $y$, we have

&c.

Next, to determine the law of connection of the quantities $A,\; A\text{'},\; A\text{'}\text{'},\; A\text{'}\text{'}\text{'},$ &c. since the last equation is to hold good whatever be the value of the variable quantity, a similar equation may be formed with the same co-efficients, and having $x\; +\; z$ for its variable ($z$ being any indefinite quantity); thus we have also

But $r^\{x+z\}\; =\; r^x\; \backslash times\; r^z$, and since it has been shown that

for the same reason

therefore the series

is equal to the product of the two series

That is, by the actual involution of the former and multiplication of the two latter,

Now the quantities $A,\; A\text{'},\; A\text{'}\text{'},$ &c. being quite independent of $x$ and $z$, the two sides of the equation can only be identical, upon the supposition that the co-efficients of like terms in each are equal; therefore, setting aside the first line of each side of the equation, because their terms are the same, and also the first term of the second line, for the same reason, let the co-efficients of the remaining terms be put equal to one another. Thus we have

Here the law of the co-efficients $A,\; A\text{'},\; A\text{'}\text{'},$ &c. is obvious,

Construct each being formed from the preceding by multiplying it by $A$, and dividing by the exponent of the power of $A$, which is thus formed. Let these values of $A^1,\; A^2,\; \backslash dots$ be now substituted in the equation

Thus we have obtained a general formula expressing a number in terms of its logarithm and the base of the system, for we must recollect that since $a\; =\; r\; -\; 1$, the quantity $A$, which is equal to

is otherwise expressed by

where $r$ denotes the base of the system.

If in the formula

we suppose $x\; =\; 1$, it becomes

an equation which contains $r$ only; but as $r$ has been all along supposed an indeterminate quantity, this equation must be identical, that is, if, instead of $A$, its value, as expressed above in terms of $r$, were substituted, the equation would become $r\; =\; r$.

Again, let us suppose that $\backslash frac\{1\}\{A\}$ is substituted instead of $x$ in the general formula; thus it becomes

Thus the quantity $r^\{\backslash frac\{1\}\{A\}\}$, whatever be the value of $r$, is evidently equal to a constant number, which, as appears from the last equation, is equal to the value of $r$ when $A\; =\; 1$. By adding together a sufficient number of terms of the last series, we find it nearly equal to

If this be denoted by $e$, we have $r^\{\backslash frac\{1\}\{A\}\}\; =\; e$, and $r\; =\; e^A$; hence, if the number $e$ be considered as the base of a logarithmic system, the quantity $A$, namely,

is the logarithm of $r$ to the base $e$. But as $r$ is not restricted here to any particular value, we may substitute $y$ instead of it, keeping in mind that $y$ denotes any number whatever, and $x$ its logarithm; thus we have $x$ the logarithm of $y$, expressed by the series

supposing that the base of the system is the number we have expressed above by $e$.

We have now found a general formula for the logarithm of any number $y$, taken according to a particular system, namely, that which has the number $e$ for its base. But it is easy from hence to find a formula, which shall apply to any system whatever. For it has been shown that the logarithms of the same number, taken according to two different systems, are to each other as the reciprocals of the

logarithms of the bases of the systems, these last logarithms being taken according to any system whatever, that is,

Construct-
tion of Lo-
garithms.

hence we find

Let the value already found for the logarithm of $y$ to base $e$ be substituted in this equation, and it becomes

which is a general formula for the logarithm of any number whatever, to the base $r$. And it is to be recollected that in the fraction $\backslash frac\{\backslash log.\; e\}\{\backslash log.\; r\}$, which is a common multiplier to the series, the logarithms are to be taken according to the same base, which however may be any number whatever.

If in the above formula we suppose $r\; =\; e$, the multiplier $\backslash frac\{\backslash log.\; e\}\{\backslash log.\; r\}$ will be unity, and the formula will become simply

as we have already remarked. Now this is the system which was adopted by Lord Napier; and although the logarithms which were computed according to this system, or upon the supposition that the radical number is 2.7182818, &c. have been called hyperbolic logarithms, because they happen to be proportional to certain hyperbolic spaces, yet, as the logarithms of every system have the same property, it is more proper to call them Napierian logarithms.

As the constant multiplier $\backslash frac\{\backslash log.\; e\}\{\backslash log.\; r\}$, which occurs in the

general formula for the logarithm of any number, is the only part of the formula which depends for its value upon the base of the system, it has been called by writers on logarithms the modulus of the system. If we suppose the logarithms taken to the base $e$, then the numerator, viz. $\backslash log.\; e$, will be unity, and the denominator will be the Napierian logarithm of $r$. If however we suppose the logarithms taken to the base $r$, then the numerator will be $\backslash log.\; e$ to base $r$; and the denominator will be unity, so that the modulus of any system whose base is $r$, is the reciprocal of the Napierian logarithm of that base; or it is the logarithm of the number $e$ (the base of the Napierian system) to the base $r$.

In the Napierian system, the modulus is unity, and hence the logarithms of this system are more easily computed than those of any other. It was, however, soon found that a system whose base should be the same as the root of the scale of the arithmetical notation, viz. the number 10, would be the most convenient of any in practice; and accordingly such a system was actually constructed by Mr Briggs. This is the only one now in common use, and is called Briggs's system, also the common system of logarithms. The modulus of this system therefore is the reciprocal of the Napierian logarithm of 10, viz. .43429448, which is the common logarithm of $e\; =\; 2.7182818$, &c. the base of the Napierian system. We shall in future denote this modulus by $M$; so that the formula expressing the common logarithm of any number $y$ will be

If the number $y$, whose logarithm is required, be very near 3 II

Construct- to unity, so that $1\; -\; y$ is a small quantity, then the loga-
tion of Lo- rithm may be found from this formula with great ease,
garithms. because the series will converge very rapidly. If, how-
ever, $1\; -\; y$ be greater than unity, the series, instead of
converging, will diverge, so as to be, in its present form,
of no use.

It may, however, be transformed into another, which
shall converge in every case, by substituting in it $\backslash sqrt[n]\{y\}$
instead of $y$, and observing that $\backslash log.\; (\backslash sqrt[n]\{y\})\; =\; \backslash frac\{\backslash log.\; y\}\{n\}$;

it thus becomes

$\backslash log.\; y\; =\; nM\; \backslash \{\; \backslash sqrt[n]\{y\}\; -\; 1\; -\; \backslash frac\{1\}\{2\}(\backslash sqrt[n]\{y\}\; -\; 1)^2\; +\; \backslash frac\{1\}\{3\}(\backslash sqrt[n]\{y\}\; -\; 1)^3\; -\; \backslash dots\; \backslash \}$
where $n$ may denote any number, positive or negative.
But whatever be the number $y$, we can always take $n$,
such, that $\backslash sqrt[n]\{y\}$ shall be as near to 1 as we please; there-
fore, by this last formula, we can always find the loga-
rithm of $y$ to any degree of accuracy.

If $n$ be taken negative, then $\backslash sqrt[n]\{y\}\; =\; \backslash frac\{1\}\{\backslash sqrt[n]\{y\}\}$, and the se-
ries for $\backslash log.\; y$ becomes, by changing the signs,

where all the terms are positive. Thus we have it in our
power to express the value of $y$, either by a series which
shall have its terms all positive, or by one which shall
have its terms alternately positive and negative: for it is
evident that if $y$ be greater than unity, $\backslash sqrt[n]\{y\}$ will also be
greater than unity, and vice versa; but the differences
will be so much the smaller as $n$ the exponent of the root
is greater; therefore $\backslash sqrt[n]\{y\}\; -\; 1$ will be positive in the first
case, and negative in the second.

Because $\backslash text\{Nap.\; log.\; \}\; 10\; =\; \backslash frac\{1\}\{M\}$, we have by the two last
formulas

also

It is evident, that by giving to $\backslash sqrt[n]\{y\}$ such a value that
$\backslash sqrt[n]\{y\}\; -\; 1$ is a fraction less than unity, we render both the
series for the value of $\backslash log.\; y$ converging; for then the
expression $1\; -\; \backslash frac\{1\}\{\backslash sqrt[n]\{y\}\}$ will also be less than unity, seeing
it is equal $\backslash frac\{\backslash sqrt[n]\{y\}\; -\; 1\}\{\backslash sqrt[n]\{y\}\}$. Therefore, in the first series, the
second and third terms (taken together as one term) con-
stitute a negative quantity; and as the same is also true
of the fourth and fifth, and so on, the amount of all the
terms after the first is a negative quantity, or one which
is to be subtracted from the first, to obtain the value of
$\backslash log.\; y$. Hence

And since, on the contrary, the terms of the second se-
ries are all positive, the amount of all the terms after the
first is a positive quantity, or one which must be added to
the first to give the value of $\backslash log.\; y$; so that

Thus we have two limits to the value of the logarithm

of $y$, which, by taking the number $n$ sufficiently great, Con-
struction of Lo-
garithms. may come as near to each other as we please.

In like manner we find two limits to the value of the
reciprocal to the modulus, viz.

It is evident that the difference between the two limits of
$\backslash log.\; y$, is

therefore, if we take either the one or the other of the two
preceding expressions for $\backslash log.\; y$, the error in excess or de-
fect is necessarily less than this quantity.

By these formulae we may depend upon having the loga-
rithm of any number true to $m$ figures, if we give to $n$
such a value that the root $\backslash sqrt[n]\{y\}$ shall have $m$ ciphers be-
tween the decimal point and the first significant figure on
the right. So that in general, as the error is the smaller
as $n$ the exponent of the root is greater, it may be neglected
when $n$ is taken indefinitely great; and this being the
case, we may conclude that either of these expressions,

is the accurate value of $\backslash log.\; y$.

The best manner of applying the preceding formula is
to take some power of the number 2 for $n$; for by doing
so, the root $\backslash sqrt[n]\{y\}$ may be found by a repetition of extrac-
tions of the square root only. It was in this way that
Briggs calculated the first logarithms; and he remarked,
that if in performing the successive extractions of the
square root, he at last obtained twice as many decimal
places as there were ciphers after the decimal point, the
integer before it being unity, then the decimal part of this
root was exactly the half of that which went before; so
that the decimal parts of the two roots were to each other
in the same proportion as their logarithms: now this is an
evident consequence of the preceding formula.

To give an example of the application of the formula,
let it be required to find the numerical value of $M$, the
modulus of the common system of logarithms, which, as
it is the reciprocal of the Napierian logarithm of 10, is

when $n$ is some very great number. Let us suppose
$n\; =\; 2^\{60\}\; =\; 8^\{20\}$; then, dividing unity by 8, and this result
again by 8, and so on, we shall, after 20 divisions, have

$\backslash frac\{1\}\{n\}$, or $\backslash frac\{1\}\{8^\{20\}\}$ equal to

Also, by extracting the square root of 10, and the square
root of this result, and so on, after performing 60 extrac-
tions we shall find $\backslash sqrt[n]\{10\}$ equal to

Therefore, $\backslash frac\{1\}\{n\}\; \backslash times\; \backslash frac\{1\}\{\backslash sqrt[n]\{10\}\; -\; 1\}$, or $M$ is equal to

As a second example, let it be required to find by the
same formula the logarithm of 3, which is nearly

$n$ being as before a very great number. Let us suppose
also in this case that $n\; =\; 2^\{60\}$; then after 60 extractions
of the square root we have $\backslash sqrt[n]\{3\}$ equal to

Therefore, taking the value of $\backslash sqrt[n]\{10\}$ as found in last example, we have

This method of computing logarithms is evidently attended with great labour, on account of the number of extractions of roots which it requires to obtain a result true to a moderate number of places of figures. But the two series which we have given serve to simplify and complete it. For, whatever be the number $y$, it is only necessary to proceed with the extractions of the square root, till we have obtained for $\backslash sqrt[n]\{y\}$ a value which is unity followed by a decimal fraction; and then $\backslash sqrt[n]\{y\}\; -\; 1$ being a fraction, its powers will also be fractions so much the smaller as their exponents are greater; thus a certain number of terms of the series will express the logarithm to as many decimal places as may be required.

There are yet other artifices by which the series $\backslash log.\; y\; =\; M\; \backslash \{\; y\; -\; 1\; -\; \backslash frac\{1\}\{2\}(y-1)^2\; +\; \backslash frac\{1\}\{3\}(y-1)^3\; -\; \backslash frac\{1\}\{4\}(y-1)^4\; +\; \backslash dots\; \backslash \}$ may be transformed into others which shall always converge, and in particular the following. Let $1\; +\; u$ be substituted in the series for $y$; then it becomes

In like manner, if $1\; -\; u$ be substituted for $y$, we have

Let the latter equation be subtracted from the former; and since $\backslash log.\; (1\; +\; u)\; -\; \backslash log.\; (1\; -\; u)$, is equal to $\backslash log.\; \backslash frac\{1\; +\; u\}\{1\; -\; u\}$; we shall have

which series, by substituting $z$ for $\backslash frac\{1\; +\; u\}\{1\; -\; u\}$, and consequently $\backslash frac\{z\; -\; 1\}\{z\; +\; 1\}$ for $u$, will be otherwise expressed thus,

which is not only simple, but has also the property of converging in every case.

As an example of the utility of this formula, we shall employ it to compute the Napierian logarithm of 2, which will be

where $A$ is put for $\backslash frac\{2\}\{3\}$, $B$ for $\backslash frac\{2\}\{3^3\}\; =\; \backslash frac\{A\}\{9\}$, $C$ for $\backslash frac\{2\}\{3^5\}\; =\; \backslash frac\{B\}\{9\}$, $D$ for $\backslash frac\{2\}\{3^7\}\; =\; \backslash frac\{C\}\{9\}$, &c. The calculation will be as follows:

Nap. log. 2 = .693147180551

Thus, by a very easy calculation, we have obtained the Napierian logarithm of 2 true to the first ten places of figures; the accurate value, as far as the 12th place, being 0.693147180560.

If this very simple process, by which we have found the logarithm of 2 (the whole of which is here actually put down), be compared with the laborious calculations which must have been performed to have found the same logarithm by the method explained in the beginning of this section, the great superiority of this method to the other, and even to the second method, by which we found the value of $M$, and the logarithm of 3, must be very apparent.

In the same manner as we have found the logarithm of 2, we may find those of 3, 5, &c. In computing the logarithm of 3, the series would converge by the powers of the fraction $\backslash frac\{3-1\}\{3+1\}\; =\; \backslash frac\{1\}\{2\}$, and in computing the logarithm of 5 it

would converge by the powers of $\backslash frac\{5-1\}\{5+1\}\; =\; \backslash frac\{2\}\{3\}$; but in each of these cases the series would converge slower, and of course the labour would be greater than in computing the logarithm of 2. And if the number whose logarithm was required was still more considerable; as, for example, 199, the series would converge so slow as to be useless.

We may however avoid this inconvenience by again transforming this last formula into another which shall express the logarithm of any number by means of a series, and a logarithm supposed to be previously known. To

effect this new transformation, let $\backslash frac\{1\; +\; u\}\{1\; -\; u\}\; =\; 1\; +\; \backslash frac\{z\}\{n\}$, and

consequently $u\; =\; \backslash frac\{z\}\{2n\; +\; z\}$, these values being substituted in

the formula, $\backslash log.\; \backslash frac\{1\; +\; u\}\{1\; -\; u\}\; =\; 2M\; \backslash left(\; u\; +\; \backslash frac\{u^3\}\{3\}\; +\; \backslash frac\{u^5\}\{5\}\; +\; \backslash frac\{u^7\}\{7\}\; +\; \backslash dots\; \backslash right)$

we have $\backslash log.\; \backslash left(\; 1\; +\; \backslash frac\{z\}\{n\}\; \backslash right)$ equal to

but $\backslash log.\; \backslash left(\; 1\; +\; \backslash frac\{z\}\{n\}\; \backslash right)\; =\; \backslash log.\; \backslash frac\{n\; +\; z\}\{n\}\; =\; \backslash log.\; (n\; +\; z)\; -\; \backslash log.\; n$,

therefore, $\backslash log.\; (n\; +\; z)\; =\; \backslash log.\; n\; +$

By the assistance of this formula, and the known properties of logarithms, we may proceed calculating the logarithm of one number from that of another as follows.

To find the Napierian logarithm of 3 from that of 2, already found. We have here $n\; =\; 2$, $z\; =\; 1$, and $\backslash frac\{z\}\{2n\; +\; z\}\; =\; \backslash frac\{1\}\{5\}$.

Therefore the logarithm of 3 is

Construction of Logarithms. where A is put for $\backslash frac\{A\}\{25\}$, B for $\backslash frac\{B\}\{25\}$, and so on.

The calculation may stand thus:

This logarithm is true to 10 decimal places, the accurate value to 12 figures being 1.098612288668, which would have been our result had the correct value of log. 2, viz. 0.693147180560 been employed.

The Napierian logarithm of 4 is immediately had from that of 2 by considering that as $4\; =\; 2^2$, therefore log. 4 = log. 2 + log. 2.

This is also true to 10 places besides the integer.

To find the Napierian logarithm of 5, from that of 4; we have $n\; =\; 4$, $z\; =\; 1$, and $\backslash frac\{z\}\{2n+1\}\; =\; \backslash frac\{1\}\{9\}$, therefore the logarithm of 5 is expressed by

where $A\; =\; \backslash frac\{2\}\{9\}$, $B\; =\; \backslash frac\{1\}\{9^2\}$, $C\; =\; \backslash frac\{1\}\{9^3\}$, &c.

The calculation.

This result is also correct to the first ten places.

The logarithm of 6 is found from those of 2 and 3, because $6\; =\; 2\; \backslash times\; 3$, therefore log. 6 = log. 2 + log. 3.

This result is correct as far as the tenth decimal place.

We might find the logarithm of 7 from that of 6, that is, from the logarithms of 3 and 2, in the same manner as we have found the logarithms of 5 and 3; but it may be more readily found from the logarithms of 2 and 5 thus.

Because $\backslash frac\{2\; \backslash times\; 5^2\}\{7^2\}\; =\; \backslash frac\{50\}\{49\}$, therefore log. 2 + 2 log. 5 =

$2\; \backslash log.\; 7\; =\; \backslash log.\; \backslash frac\{50\}\{49\}$, and consequently

Now the logarithm of $\backslash frac\{50\}\{49\}$ may be readily obtained from

For, substituting $\backslash frac\{50\}\{49\}$ for $z$, the formula gives

where $A\; =\; \backslash frac\{2\}\{9\; \backslash cdot\; 11\}$, $B\; =\; \backslash frac\{A\}\{9^2\; \backslash cdot\; 11^2\}$, $C\; =\; \backslash frac\{B\}\{9^4\; \backslash cdot\; 11^4\}$, &c. This series converges with great rapidity, and a few of its terms will be sufficient to give the logarithm of 7, as appears from the following operation.

This logarithm, like those we found before, is correct in the first ten decimal places.

The logarithms of 8, 9, and 10 are immediately obtained from those of 2, 3, and 5, as follows:

Thus by a few calculations we have found the Napierian logarithms of the first ten numbers, each true to ten decimal places; and since the Napierian logarithm of 10 is now known, the modulus of the common system, which is the reciprocal of that logarithm, will also be known, and will be

The common logarithms of the first ten numbers may now be found from the Napierian logarithms by multiplying each of the latter by the modulus, or dividing by its reciprocal, that is, by the Napierian logarithm of 10. And as the modulus of the common system is so important an element in the theory of logarithms, we shall give its value, together with that of its reciprocal, as far as the 30th decimal place.

The formulæ already given are sufficient for finding the logarithms of all numbers whatever throughout the table; but there are yet others which may often be applied with great advantage, and we shall now investigate some of these.

If we now suppose

so that $\backslash frac\{z-1\}\{z+1\}\; =\; \backslash frac\{1\}\{2n^2-1\}$, then the formula becomes

But $\backslash log.\; \backslash frac\{n^2\}\{(n-1)(n+1)\}\; =\; 2\; \backslash log.\; n\; -\; \backslash log.\; (n-1)\; -\; \backslash log.\; (n+1)$, therefore, putting $N$ for the series

we have this formula,

$2\; \backslash log.\; n\; -\; \backslash log.\; (n-1)\; -\; \backslash log.\; (n+1)\; =\; N$; and hence, as often as we have the logarithms of any two of three numbers whose common difference is unity, the logarithm of the remaining number may be found. Example: Given $\backslash log.\; 9\; =\; 0.95424250943$, $\backslash log.\; 10\; =\; 1$; to find the common logarithm of 11.

Here we have $n\; =\; 10$, so that the formula gives in this case $2\; \backslash log.\; 10\; -\; \backslash log.\; 9\; -\; \backslash log.\; 11\; =\; N$, and hence we have

$M$ being 43429448190.

Calculation of $N$.

Here the series expressed by $N$ converges very fast, so that two of its terms are sufficient to give the logarithm true to 10 places of decimals. But the logarithm of 11 may be expressed by the logarithms of smaller numbers, and a series which converges still more rapidly, by the following artifice, which will apply also to some other numbers. Because the numbers 98, 99, and 100 are the products of numbers, the greatest of which is 11, for $98\; =\; 2\; \backslash times\; 7^2$, $99\; =\; 9\; \backslash times\; 11$, and $100\; =\; 10\; \backslash times\; 10$, it follows that if we have an equation composed of terms which are the logarithms of these three numbers, it may be resolved into another, the terms of which shall be the logarithms of the number 11 and other smaller numbers. Now by the preceding formula, if we put 99 for $n$, we have

that is, substituting $\backslash log.\; 9\; +\; \backslash log.\; 11$ for $\backslash log.\; 99$, $\backslash log.\; 2\; +\; 2\; \backslash log.\; 7$ for $\backslash log.\; 98$, and $2\; \backslash log.\; 10$ for $\backslash log.\; 100$,

and hence by transposition, &c.

and in this equation

The first term alone of this series is sufficient to give the logarithm of 11 true to 14 places.

Another formula, by which the logarithm of a number is expressed by the logarithms of other numbers and a series, may be found as follows.

Resuming the formula

Let these values of $z$, and $\backslash frac\{z-1\}\{z+1\}$, be substituted in the formula, and it becomes

But the quantity on the left-hand side of this equation is manifestly equal to $2\; \backslash log.\; (n-1)\; +\; \backslash log.\; (n+2)\; -\; \backslash log.\; (n-2)\; -\; 2\; \backslash log.\; (n+1)$; therefore, putting $P$ for the series,

we have this formula,

$\backslash log.\; (n+2)\; +\; 2\; \backslash log.\; (n-1)\; -\; \backslash log.\; (n-2)\; -\; 2\; \backslash log.\; (n+1)\; =\; P$. By this formula we may find, with great facility, the logarithm of any one of the four numbers $n-2$, $n-1$, $n+1$, $n+2$, having the logarithms of the other three. We may also employ it in the calculation of logarithms, as in the following example. Let the numbers 5, 6, 7, 8, be substituted successively in the formula; then, observing that $\backslash log.\; 6\; =\; \backslash log.\; 2\; +\; \backslash log.\; 3$, and $\backslash log.\; 8\; =\; 3\; \backslash log.\; 2$, we have these four equations,

Let $\backslash log.\; 2$, $\backslash log.\; 3$, $\backslash log.\; 5$, and $\backslash log.\; 7$, be now considered as

Construct four unknown quantities, and by resolving those equations in the usual manner, the logarithms may be determined. Construct. of Logarithms.

Resuming once more the formula

let $\backslash frac\{n^2(n+5)(n-5)\}\{(n+3)(n-3)(n+4)(n-4)\}$ be substituted in it

instead of $z$, then, by this substitution, $\backslash frac\{z-1\}\{z+1\}$ will become

$\backslash frac\{-72\}\{n^4\; -\; 25n^2\; +\; 72\}$, and the formula will be transformed to

Hence, putting the latter side of this equation equal to $Q$, we have this formula,

which may be applied in the calculation of logarithms in the same manner as the former.

But when it is required to find the logarithm of a higher number, as, for example, 1231, we may proceed as follows.

Again, $\backslash log.\; 1230\; =\; \backslash log.\; 2\; +\; \backslash log.\; 5\; +\; \backslash log.\; 123$, and $\backslash log.$

Thus the logarithm of the proposed number is expressed by the logarithms of 2, 3, 5, and the logarithms of

$1\; +\; \backslash frac\{1\}\{40\}$, $1\; +\; \backslash frac\{1\}\{1230\}$, all of which may be easily found by the formulae already delivered.

It may be proper here briefly to notice some new views on the doctrine of logarithms, which have been advanced at considerable length by Mr Graves in the Philosophical Transactions for 1829. The discovery made by Poisson and Poinset, in their researches on angular sections, of errors in trigonometrical formulae usually deemed complete, drew Mr Graves's attention to what he considered analogous defects in logarithmic series. He accordingly professes, in the paper cited, to exhibit in an amended form two fundamental developments, especially that of the one we have given above of the equation $y\; =\; r^x$; but the new formulae are unfortunately exhibited in such an abstruse and indefinite form, as to be quite unintelligible to most readers, and our limits will not admit of introducing them here with a sufficient explanation. The chief objection made to the ordinary series is, that it only gives one value to $y$ for each value of $x$; whereas, when $x$ is a rational fraction in its lowest terms, $y$ should have as many values of some sort as there are units in the denominator of $x$. Thus, when $x\; =\; \backslash frac\{1\}\{2\}$ and $r\; =\; 10$, the value of $y$ being $\backslash pm\; \backslash sqrt\{10\}$, may be $-3.162277$ as well as $+3.162277$; but it is surely never meant that $y$ should ever have more

than two possible values, or indeed more than one such value, when the denominator of $x$ is odd. Construct. of Logarithms.

Mr Graves considers the principles employed in this inquiry as presenting a solution of many difficulties, and illustrating peculiarities in the theory of logarithms of negative quantities; and, when applied to geometry, as furnishing the means of tracing the form and developing the properties of curves whose equations involve exponential quantities. He also states that, by their means, various differential and other formulae usually exhibited in treatises on logarithms may be rendered complete. He combats the opinion that equations which are numerically false may yet be analytically true; and explains the difficulty by reverting to the limitations inherent in the hypothesis on which the developments are founded. In opposition to John Bernoulli and D'Alembert, he maintains that the logarithms of negative numbers are not in general the same as those of positive ones; and hence infers that negative numbers have occasionally even real logarithms. But several objectionable things in Mr Graves's paper have been pointed out by Mr Peacock, in an able article on analysis in the Third Report of the British Association, page 266; and, indeed, until Mr Graves chooses to put his new views in a more definite and intelligible shape, and to express himself in ordinary mathematical language, we suspect his speculations are not likely to meet with a favourable reception.

In addition to the works already mentioned on the theory of logarithms, the following may be consulted, viz.

James Gregory, Vera Circuli et Hyperbole Quadratura, 1667.

James Gregory, Exercitationes Geometricæ, 1668.

James Gregory, in Commercium Epistolicum, 1712.

Mercator, Logarithmotechnia, 1668; and Philosophical Transactions of same year.

Brouncker and Wallis, Phil. Trans. 1668.

Barrow, Lectiones Geometricæ, 1674.

Halley, Phil. Trans. 1695, 1696.

Jones, Synopsis Matheseos, 1706.

Craig, Phil. Trans. 1710.

Sir Isaac Newton, Commercium Epistolicum.

Sir Isaac Newton, Method of Fluxions, 1736.

Cotes, Phil. Trans. 1714.

Cotes, Harmonia Mensurarum, 1722.

Long, Phil. Trans. 1714.

Brooke Taylor, Phil. Trans. No. 352.

Dodson, Phil. Trans. 1753.

Maseres, Trigonometry, 1760.

Maseres, Scriptores Logarithmici, which is a valuable collection of the more interesting and scarce tracts on logarithms. It is in six vols. 4to, published from 1791 to 1807.

Waring, Phil. Trans. 1779.

Hellins, Phil. Trans. 1780-1796.

Lagrange, Théorie des Fonctions.

Laplace, Journal de l'École Polytechnique, 1809.

Wallace on the Conic Sections, in Transactions of the Royal Society of Edinb. vol. vi.

Lavernède, Annales des Mathématiques, 1811.

Vincent, Annales des Mathématiques, 1824, 1825.

To most of the larger sets of tables is prefixed an introduction on the theory and construction of logarithms.

The following short article may be fitly introduced here as an addition to the foregoing treatise. It was written by the late Sir John Leslie for the Supplement to the former editions of this work, in which it was inserted under the word INTERPOLATION, in the Addenda. Easy mode of computing logarithms by interpolation.

Few disquisitions in modern mathematics have greater practical utility than those concerning the extension and

Construction of the terms of numerical progressions. But the modes of solution generally proposed, however ingenious and refined, are often too complex for ordinary purposes. We shall therefore point out a procedure of extreme simplicity and most ready application. It will embrace any system of numbers, but seems peculiarly fitted for the computation of logarithms.

Napier first published in 1614 his canon Mirificus Logarithmorum, comprised in a very thin and small quarto volume, exhibiting, as far as seven places, the logarithms only of the sines and tangents to every minute of the quadrant; those of the ordinary numbers being left to be deduced from the nearest sines or tangents. In the summer following, Briggs, then professor of geometry in Gresham College at London, who was enchanted by that noble discovery, paid a visit to its illustrious author at Merchiston, in the vicinity of Edinburgh. During an agreeable stay of a month, it was concerted between them to change the natural system of logarithms into another of a more artificial form, but adapted to our denary scale of notation, the labour of the calculation, however, being devolved upon Briggs, as younger, and enjoying robust health. Next season, Briggs performed a second journey to Edinburgh, and showed to Napier a table which he had computed of the new logarithms for the first chiliad or thousand of the series of natural numbers. But the great inventor was now fast declining in years, and expired on the 3d of April 1618. Briggs, employing most skilfully all the abbreviations which ingenuity could devise, prosecuted his most arduous task with such vigour and active perseverance, as to compute, in the space of seven years, to fourteen places of figures, not only the logarithms of the sines and tangents of every degree and centesimal minute of the quadrant, but also the logarithms of thirty chiliads of numbers. These tables were printed in 1624, in a folio volume, entitled Arithmetica Logarithmica. In his anxiety, however, to bring out this stupendous work, Briggs contented himself with calculating the logarithms from unit to 20,000, and from 90,000 to 100,000, leaving the computation of the seventy intervening chiliads to be afterwards supplied. The very ingenious author annexed a full explanation of his mode of framing the tables, grounded chiefly on the consideration of differences, and he thus traced the first steps of that important theory. But he likewise gave instructions for the readier filling up of the intermediate logarithms, abridging the toil of calculation no doubt, yet detailing a procedure sufficiently irksome and complicated. "To encourage some skilful persons to perform this task, he offered to furnish them with paper he had by him, ready prepared, and divided into columns proper for that purpose, as likewise to inform them at what part to begin, that they might not interfere one with another; and promised, when the whole was finished, to endeavour to procure a new edition of the work so completed." This slender boon seems not to have tempted the mathematicians in England. In Holland, however, Adrian Vlacq, chancing to meet two years afterwards with a copy of the work, and prompted only by his patriotic zeal, had the resolution to revise and compute the whole canon, reducing it from fourteen to ten places of figures, which he printed in folio at Gouda, as early as 1628. The same able calculator, only five years afterwards, published, likewise in folio, at Gouda, a very extensive system of logarithmic sines and tangents, to every ten seconds of the quadrant, having restored the sexagesimal subdivision, which Briggs had partly changed into a centesimal one. These two volumes may be deemed a precious thesaurus of logarithms, from which succeeding compilers have drawn very liberally. They form the basis of the Tables published by Vega at Leipsic in 1794, which, till lately, were the best and completest extant.

But though the Tables of Vlacq, carried only to ten places of figures, are sufficiently accurate for every ordinary purpose, and even for the most delicate calculations in astronomy, yet many persons have often regretted that the original system of Briggs was never completed. The celebrated Legendre has employed that table, imperfect as it is, in some of his most refined numerical investigations. It is well known that Mr Baron Maseres devoted a considerable portion of his time and of his fortune to the republication of the works of the early writers on logarithms. In the course of this extensive undertaking, he entertained some thoughts of giving a new edition of the Arithmetica Logarithmica, and expressed an earnest wish that the vacant chiliads were filled up. To promote the liberal designs of the baron, the author of this article was induced to bestow some reflection on the subject, and a very simple mode occurred to him, which would have reduced the labour of computing those logarithms to little more than the trouble of mere transcription. But the object of completing the canon was deferred for a time, and afterwards gradually forgotten. The method of interpolation then proposed seems, however, to deserve notice on account of its great simplicity, and its ready application, not only to the immediate object, but to other questions of a similar nature. We shall, therefore, now state the principle, and illustrate its application by a few examples.

The square root of the quantity $a^2\; +\; 1$ is evidently expressed by the continued fraction $a\; +\; \backslash frac\{1\}\{2a\; +\; \backslash frac\{1\}\{2a\; +\; \backslash dots\}\}$

If two terms only of the fractional part be taken, the expression will become $\backslash sqrt\{a^2\; +\; 1\}\; =\; \backslash frac\{4a^2\; +\; 3a\}\{4a^2\; +\; 1\}$, and consequently $\backslash sqrt\{\backslash left(\backslash frac\{a^2\; +\; 1\}\{a^2\}\backslash right)\}$ or $\backslash sqrt\{\backslash left(\backslash frac\{2a^2\; +\; 2\}\{2a^2\}\backslash right)\}\; =\; \backslash frac\{4a^2\; +\; 3\}\{4a^2\; +\; 1\}$, a very near approximation. Put $b\; =\; 2a^2$, and by substitution $\backslash sqrt\{\backslash left(\backslash frac\{b+2\}\{b\}\backslash right)\}\; =\; \backslash frac\{2b+3\}\{2b+1\}$; wherefore $\backslash frac\{1\}\{2\}\; \backslash log\; \backslash frac\{b+2\}\{b\}\; =\; \backslash log\; \backslash frac\{2b+3\}\{2b+1\}$.

Hence half the differences of the alternate logarithms of the series $b,\; b+1,\; b+2,\; \backslash dots$ added to the logarithm of $2b+1$, must give the logarithm of $2b+3$. By this simple process, then, any table of logarithms is carried to double its actual extent through all the odd numbers, those of the even ones being found by the mere addition of the logarithm of 2.

To find the limits of approximation, let three terms of the fractional series be taken, and $\backslash sqrt\{\backslash left(\backslash frac\{a^2\; +\; 1\}\{a^2\}\backslash right)\}\; =\; \backslash frac\{8a^4\; +\; 8a^2\; +\; 1\}\{8a^4\; +\; 4a^2\}$, or $\backslash sqrt\{\backslash left(\backslash frac\{b+2\}\{b\}\backslash right)\}\; =\; \backslash frac\{2b^2\; +\; 4b\; +\; 1\}\{2b^2\; +\; 2b\}\; =\; \backslash frac\{2b+3\}\{2b+1\}\; \backslash left(\; \backslash frac\{4b^3\; +\; 10b^2\; +\; 6b\; +\; 1\}\{4b^3\; +\; 10b^2\; +\; 6b\}\; \backslash right)\; =\; \backslash frac\{2b+3\}\{2b+1\}\; \backslash left(\; 1\; +\; \backslash frac\{1\}\{(2b+3)(2b+1)b\}\; \backslash right)$. Whence $\backslash frac\{1\}\{2\}\; \backslash log\; \backslash left(\; \backslash frac\{b+2\}\{b\}\; \backslash right)\; =\; \backslash log\; \backslash frac\{2b+3\}\{2b+1\}\; +\; \backslash log\; \backslash left(\; 1\; +\; \backslash frac\{1\}\{(2b+3)(2b+1)b\}\; \backslash right)$; and, therefore, since this last quantity exceeds unit only by a very minute difference, $\backslash frac\{1\}\{2\}\; \backslash log\; \backslash frac\{b+2\}\{b\}\; =\; \backslash log\; \backslash frac\{2b+3\}\{2b+1\}\; +\; \backslash frac\{M\}\{(2b+3)(2b+1)b\}$, where $M$ denotes the modulus of the system. If the number $2b+3$ or $2b+1$ be ex-

Construction of Logarithms. pressed by $N$, this small correction will amount but to $\backslash frac\{M\}\{\backslash frac\{1\}\{2\}N^2\}$ or $\backslash frac\{2M\}\{N^2\}$. Consequently $\backslash log\; \backslash frac\{2b+3\}\{2b+1\}\; =\; \backslash frac\{1\}\{2\}\; \backslash log\; \backslash frac\{b+2\}\{b\}\; -\; \backslash frac\{2M\}\{N^2\}$. It may hence be computed, that the correction

on the first approximation will only reach unit in the last figure for the logarithms of numbers under 206 in tables of seven places, for those under 2055 in tables of ten places, and for the logarithms of numbers under 44286 in tables extending to fourteen places.

The corrections required in Briggs's Tables of fourteen places will therefore correspond to these limits:

Suppose it were required to find the logarithms of the odd numbers above 300, to seven places of decimals. Assuming the logarithms of the series of the halves of the intermediate even numbers, let the differences between their alternate terms be taken, and then bisected.

Hence the logarithms of the doubles are formed by the mere addition of these halves.

In this manner the operation may be continued; but, to prevent any accumulation of errors, the logarithms of the composite numbers should serve as standards, being formed by the addition of the logarithms of their several factors. The logarithms of the intermediate even numbers 302, 304, 306, 308, and 310, are easily determined by adding .3010300 to the logarithms of 151, 152, 153, 154, and 155.

To extend the process a little farther, let Vlaccq's logarithms be computed for the numbers above 4000.

Whence are derived,

Again, to compute the logarithms in Briggs's Canon.

The additive parts here consist of those half differences diminished by 11 and the last one by 10, since the numbers now approach to the limit 20557.

The first mode of interpolating, thus derived from the nature of logarithms, and so commodious for their computation, might likewise be deduced from general considerations. Let represent any series of numbers. If they advance regularly and slowly, their first differences, may be viewed as constituting an arithmetical progression. Wherefore the sum of the extremes will be equal to that of the mean terms, or $(E-D)\; +\; (B-A)\; =\; (D-C)\; +\; (C-B)$, that is, $E-A\; =\; 2D-2B$, and therefore $D-B\; =\; \backslash frac\{E-A\}\{2\}$, whence $D\; =\; B\; +\; \backslash frac\{E-A\}\{2\}$. Applying

this to the logarithms of eight places of figures, let represent the logarithms of 500, 501, 502, 503, 504, &c. then $\backslash log.\; 503\; =\; \backslash log.\; 501\; +\; \backslash frac\{\backslash log.\; 504\; -\; \backslash log.\; 500\}\{2\}$
$=\; \backslash log.\; 501\; +\; \backslash frac\{\backslash log.\; 252\; -\; \backslash log.\; 250\}\{2\}\; =\; 2.69983773\; +\; \backslash frac\{1\}\{2\}$
$(2.40140054\; -\; 2.39794001)\; =\; 2.79156799$; the halves of the differences of the alternate logarithms of 250, 251, 252, 253, &c. being thus taken, as before, to compose by their additions the logarithms of the odd numbers 503, 505, 507, &c.