I.—HISTORY OF LOGARITHMS.

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:

| Geom. Prog. | Arith. Prog. | |------------|-------------| | 1 | 0 | | 2 | 1 | | 4 | 2 | | 8 | 3 | | 16 | 4 | | 32 | 5 | | 64 | 6 | | 128 | 7 | | 256 | 8 | | 512 | 9 | | 1024 | 10 | | 2048 | 11 | | 4096 | 12 | | &c. | &c. |

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 $10^4 = 16 \times 64$; 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 $1024 \div 16 = 64$, 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 anything 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 equably, 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, praemissa Demonstrazione legitima Ortes Logarithmorum corumque Usus, &c.; and in the following year he published a supplement to this work. In the 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 $2^{3203851}$. 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 Chilias 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 Robologia, 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 History 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 Chilias 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 Vlacq or Flack of Gouda, in Holland, completed the intermediate 70 chilads, 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 Vlacq, who in the same year printed his own Trigonometria Artificialis, sive Magnum Canon Triangulorum Logarithmicos ad Decadas Secundorum Sexpulorum 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 computors 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 millième, &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 Montpellier, 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.

Shortrede's Tables.—These tables contain the logarithms to numbers from 1 to 120,000, and numbers to logarithms from '0 to '100000; tables with centesimal and decimal arguments for finding logarithms and antilogarithms, as far as sixteen and twenty-five places; tables to five places for finding the logarithms of the sums and differences of antilogarithms, &c.; tables of natural sines, cosines, &c., to every second of arc in the quadrant; tables containing angles which every point and quarter-point of the compass makes with the meridian in points and degrees; tables of logarithmic sines, cosines, tangents, &c., to every second of the circle, with arguments in space and time.

II.—NATURE OF LOGARITHMS AND THEIR CONSTRUCTION.

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 2·718282, by which it happened that the logarithm of 10 was 2·302585; 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 2·718282, as in Napier's first system; and hence the logarithms of the terms of the geometrical progression

\[1, 10, 100, 1000, 10,000, \ldots\]

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

\[0, 1, 2, 3, 4, \ldots\] 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 \( \sqrt{1 \times 10} = \sqrt{10} \); and the arithmetical mean will be half the sum of the logarithms 0 and 1, which is 0.5; therefore the logarithm of \( \sqrt{10} \) 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 \( \sqrt{10} \), 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 \( \sqrt{\frac{1 + 0.5}{2}} = 0.75 \), the logarithm of 5.623413.

Third step.—We have now obtained two numbers, namely, \( \sqrt{10} \) 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, viz. \( \sqrt{\frac{1 + 0.75}{2}} = 0.625 \), 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:

| Numbers | Logarithms | |---------|------------| | A | 1.000000 | | B | 10.000000 | | C | \( \sqrt{AB} = \sqrt{10} \) | | D | \( \sqrt{BC} = 5.623413 \) | | E | \( \sqrt{CD} = 4.216964 \) |

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 0.6875000 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 Wlaq 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 to illustrate by actual computation.

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 0.6875000, 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 \( 4 = 2 \times 2 \), the logarithm of 4 will be the logarithm of 2 added to itself, or will be twice the logarithm of 2. Again, because \( 5 \times 10 = 50 \), 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 = \( 2 \times 4 \), of 16 = \( 2 \times 8 \), of 25 = \( 5 \times 5 \), 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 301,930 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, 301,930, 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 λογος ἀριθμος, 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 301,930 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 301,930, 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 $A \times B$ to 1, is compounded of the ratio of $A$ to 1 and of $B$ to 1; and consequently the logarithm of the ratio of $A \times B$ 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 $A \times B$ will be the sum of the logarithms of $A$ and $B$. Now, $\log(A \times B) = \log A + \log B$, therefore, $\log B = \log(A \times B) - \log A$. Let $\frac{C}{D}$ be substituted for $B$, and $D$ for $A$, then (because $A \times B = D \times \frac{C}{D} = C$) we have $\log \frac{C}{D} = \log C - \log D$.

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 $4R^2 + 7R^4 + 1R^3 + 5R^5 + 0R^1 + 9R^0$, which is Let \( n \) express any number whatever, then raising both sides of the equation \( a = r^A \) to the \( n \)th power, we have \( a^n = (r^A)^n = r^{An} \); but here \( An \) 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 \( b \) be a negative we would have \( a = r^{-A} = \frac{1}{r^A} \), a proper 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 \( r' \) denote bases of two different systems; and let \( A \) be the logarithm of a number, \( a \), taken according to the first of these, and \( A' \) its logarithm according to the second. Then, because \( a = r^A \), and \( a = r'^{A'} \), it follows that

\[ r^A = r'^{A'}, \text{ and } r = r'^{A}. \]

Let us now suppose that \( r'' \) is the base of a third system of logarithms, and \( R \) and \( R' \), Nature of Logarithms.

the logarithms of \( r \) and \( r' \), taken according to this third system; then, because

\[ r^R = r, \quad r'^R = r'; \]

we have \( r^R r'^R = r'^R r^R \); therefore \( r'^R = r^R \), and \( r = r'^R \); but we have already found \( r = r'^A \), therefore \( r'^A = r^R \), and consequently

\[ \frac{A'}{A} = \frac{R}{R'}, \quad \text{and } A : A' :: R : R'. \]

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

\[ r^A = a, \quad r^B = b; \]

we have \( r^{AB} = a^B, \quad r^{AB} = b^A \);

therefore \( a^B = b^A \), and \( a = b^A \). Now as \( r \) is not found in this equation, the value of the fraction \( \frac{A}{B} \) 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,

\[ y = r^x. \]

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

\[ y = 1 + x + \frac{x(x-1)}{1 \cdot 2} a^2 + \frac{x(x-1)(x-2)}{1 \cdot 2 \cdot 3} a^3 + \frac{x(x-1)(x-2)(x-3)}{1 \cdot 2 \cdot 3 \cdot 4} a^4 + \cdots \]

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:

\[ 1 = 1, \]

\[ x = a + a^2 x, \]

\[ \frac{x(x-1)}{1 \cdot 2} a^2 = -\frac{a^2}{2} x + \frac{a^2}{2} x^2, \]

\[ \frac{x(x-1)(x-2)}{1 \cdot 2 \cdot 3} a^3 = +\frac{a^3}{3} x - \frac{a^3}{2} x^2 + \frac{a^3}{6} x^3, \]

\[ \frac{x(x-1)(x-2)(x-3)}{1 \cdot 2 \cdot 3 \cdot 4} a^4 = -\frac{a^4}{4} x + \frac{11a^4}{24} x^2 - \frac{a^4}{4} x^3 + \frac{a^4}{24} x^4, \]

\[ \text{etc.} \]

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

\[ y = 1 + \left( a - \frac{a^2}{2} + \frac{a^3}{3} - \frac{a^4}{4} + \cdots \right) x + \left( \frac{a^2}{2} - \frac{a^3}{3} + \frac{11a^4}{24} - \cdots \right) x^2 + \left( \frac{a^3}{6} - \frac{a^4}{4} + \cdots \right) x^3 + \left( \frac{a^4}{24} - \cdots \right) x^4 + \cdots \]

Put \( A = a - \frac{a^2}{2} + \frac{a^3}{3} - \frac{a^4}{4} + \cdots \),

\[ A' = \frac{a^2}{2} - \frac{a^3}{3} + \frac{11a^4}{24} - \cdots, \]

\[ A'' = \frac{a^3}{6} - \frac{a^4}{4} + \cdots, \]

\[ A''' = \frac{a^4}{24} - \cdots, \]

\[ \text{etc.} \]

then \( r^x = 1 + Ax + A'x^2 + A''x^3 + A'''x^4 + \cdots \).

Next, to determine the law of connection of the quantities \( A, A', A'', A''' \), etc., 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

\[ r^{x+z} = 1 + A(x+z) + A'(x+z)^2 + A''(x+z)^3 + \cdots. \]

But \( r^{x+z} = r^x \times r^z \), and since it has been shown that

\[ r^x = 1 + Ax + A'x^2 + A''x^3 + A'''x^4 + \cdots, \]

for the same reason

\[ r^z = 1 + Az + A'z^2 + A''z^3 + A'''z^4 + \cdots, \]

therefore the series

\[ 1 + A(x+z) + A'(x+z)^2 + A''(x+z)^3 + A'''(x+z)^4 + \cdots, \]

is equal to the product of the two series

\[ 1 + Ax + A'x^2 + A''x^3 + A'''x^4 + \cdots, \]

\[ 1 + Az + A'z^2 + A''z^3 + A'''z^4 + \cdots. \]

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

\[ 1 + Ax + A'x^2 + A''x^3 + A'''x^4 + \cdots, \]

\[ + Az + A'z^2 + A''z^3 + A'''z^4 + \cdots, \]

\[ = \left\{ \begin{array}{c} 1 + A(x+z) + A'(x+z)^2 + A''(x+z)^3 + A'''(x+z)^4 + \cdots \\ + A(x+z) + A'(x+z)^2 + A''(x+z)^3 + A'''(x+z)^4 + \cdots \end{array} \right\} \]

Now the quantities \( A, A', A'', A''' \), etc., 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 reasons, let the co-efficients of the remaining terms be put equal to one another. Thus we have

\[ A = 2A', \]

\[ A' = \frac{A^2}{1 \cdot 2}, \]

\[ A'' = \frac{A^3}{1 \cdot 2 \cdot 3}, \]

\[ A''' = \frac{A^4}{1 \cdot 2 \cdot 3 \cdot 4}, \]

\[ \text{etc.} \]

Here the law of the co-efficients \( A, A', A'', A''' \), etc., is obvious, Construc. each being formed from the preceding by multiplying it by tion of Lo. A, and dividing by the exponent of the power of A, which logarithms, is thus formed. Let these values of \( A^1, A^2, \ldots \) be now substituted in the equation

\[ y = r^x = 1 + Ax + A^2 x^2 + A^3 x^3 + \ldots \]

and it becomes

\[ y = 1 + Ax + \frac{A^2}{1 \cdot 2} x^2 + \frac{A^3}{1 \cdot 2 \cdot 3} x^3 + \frac{A^4}{1 \cdot 2 \cdot 3 \cdot 4} x^4 + \ldots \]

Thus we have obtained a general formula expressing a number in terms of its logarithm and the base of the sys- tem; for we must recollect that since \( a = r - 1 \), the quan- tity \( A \), which is equal to

\[ a = \frac{a^1}{2} + \frac{a^2}{3} + \frac{a^3}{4} + \frac{a^4}{5} + \ldots \]

is otherwise expressed by

\[ r - 1 = \frac{(r-1)^2}{2} + \frac{(r-1)^3}{3} + \frac{(r-1)^4}{4} + \frac{(r-1)^5}{5} + \ldots \]

where \( r \) denotes the base of the system.

If in the formula

\[ r^x = 1 + Ax + \frac{A^2}{1 \cdot 2} x^2 + \frac{A^3}{1 \cdot 2 \cdot 3} x^3 + \frac{A^4}{1 \cdot 2 \cdot 3 \cdot 4} x^4 + \ldots \]

we suppose \( x = 1 \), it becomes

\[ r = 1 + A + \frac{A^2}{1 \cdot 2} + \frac{A^3}{1 \cdot 2 \cdot 3} + \frac{A^4}{1 \cdot 2 \cdot 3 \cdot 4} + \ldots \]

an equation which contains only \( r \); 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 ex- pressed above in terms of \( r \), were substituted, the equation would become \( r = r \).

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

\[ \frac{1}{r^x} = 1 + \frac{1}{1 \cdot 2} + \frac{1}{1 \cdot 2 \cdot 3} + \frac{1}{1 \cdot 2 \cdot 3 \cdot 4} + \ldots \]

Thus the quantity \( r^{-x} \), whatever be the value of \( r \), is evi- dently 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

\[ 2.7182818284590452353602874. \]

If this be denoted by \( e \), we have \( r^e = e \); and \( r = e^A \); hence, if the number \( e \) be considered as the base of a logarithmic system, the quantity \( A \), namely,

\[ r - 1 = \frac{(r-1)^2}{2} + \frac{(r-1)^3}{3} + \frac{(r-1)^4}{4} + \frac{(r-1)^5}{5} + \ldots \]

is the logarithm of \( r \) to the base \( e \). But as \( r \) is not restrict- ed 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 loga- rithm of \( y \), expressed by the series

\[ y - 1 = \frac{(y-1)^2}{2} + \frac{(y-1)^3}{3} + \frac{(y-1)^4}{4} + \frac{(y-1)^5}{5} + \ldots \]

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 lo- garithms of the same number, taken according to two dif- ferent systems, are to each other as the reciprocals of the logarithms of the bases of the systems, these last loga- logarithms being taken according to any system whatever, that is,

\[ \log_e y : \log_r y :: \frac{1}{\log_e} : \frac{1}{\log_r}; \]

hence we find

\[ \log_e y : \log_r y = \frac{\log_e}{\log_r} \times \log_y \text{ to base } e. \]

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

\[ \log_y = \frac{\log_e}{\log_r} \left[ y - 1 - \frac{(y-1)^2}{2} + \frac{(y-1)^3}{3} - \frac{(y-1)^4}{4} + \ldots \right] \]

which is a general formula for the logarithm of any num- ber whatever, to the base \( r \). And it is to be recollected that in the fraction \( \frac{\log_e}{\log_r} \), 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 \( \frac{\log_e}{\log_r} \) will be unity, and the formula will become simply

\[ \log_y = y - 1 - \frac{(y-1)^2}{2} + \frac{(y-1)^3}{3} - \frac{(y-1)^4}{4} + \ldots \]

as we have already remarked. Now this is the system which was adopted by Lord Napier; and although the lo- garithms 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 \( \frac{\log_e}{\log_r} \) 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. \( \log_e \), will be unity, and the denominator will be the Na- pierian logarithm of \( r \). If however we suppose the lo- garithms taken to the base \( r \), then the numerator will be \( \log_r \) to base \( e \); and the denominator will be unity, so that the modulus of any system whose base is \( r \) is the re- ciprocal 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 com- puted 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 num- ber 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 lo- garithms. The modulus of this system therefore is the reci- procal 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

\[ \log_y = M \left[ 1 - y - \frac{(1-y)^2}{2} + \frac{(1-y)^3}{3} - \frac{(1-y)^4}{4} + \ldots \right] \]

If the number \( y \), whose logarithm is required, be very near LoGARITHMS.

Construction of Logarithms.

To unity, so that \(1 - y\) is a small quantity, then the logarithm may be found from this formula with great ease, because the series will converge very rapidly. If, however, \(1 - y\) 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 \(n\sqrt{y}\) instead of \(y\), and observing that \(\log\left(n\sqrt{y}\right) = \frac{\log y}{n}\), it thus becomes

\[\log y = nM\left\{n\sqrt{y} \bigg| - \frac{1}{2} \left(n\sqrt{y} - 1\right)^2 + \frac{1}{3} \left(n\sqrt{y} - 1\right)^3 - \text{&c.}\right\},\]

where \(n\) may denote any number, positive or negative. But whatever be the number \(y\), we can always take \(n\) such that \(n\sqrt{y}\) shall be as near to 1 as we please; therefore, by this last formula, we can always find the logarithm of \(y\) to any degree of accuracy.

If \(n\) be taken negative, then \(n\sqrt{y} = \frac{1}{n\sqrt{y}}\), and the series for log \(y\) becomes, by changing the signs,

\[\log y = nM\left\{1 - \frac{1}{n\sqrt{y}} + \frac{1}{2} \left(1 - \frac{1}{n\sqrt{y}}\right)^2 + \frac{1}{3} \left(1 - \frac{1}{n\sqrt{y}}\right)^3 + \text{&c.}\right\},\]

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, \(n\sqrt{y}\) also 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 \(n\sqrt{y} - 1\) will be positive in the first case, and negative in the second.

Because Nap. log. 10 = \(\frac{1}{M}\), we have by the two last formulas

\[\frac{1}{M} = n\left\{n\sqrt{10} - 1 - \frac{1}{2} \left(n\sqrt{10} - 1\right)^2 + \frac{1}{3} \left(n\sqrt{10} - 1\right)^3 + \text{&c.}\right\},\]

also

\[\frac{1}{M} = n\left\{1 - \frac{1}{n\sqrt{10}} + \frac{1}{2} \left(1 - \frac{1}{n\sqrt{10}}\right)^2 + \frac{1}{3} \left(1 - \frac{1}{n\sqrt{10}}\right)^3 + \text{&c.}\right\}.\]

It is evident, that by giving to \(n\sqrt{y}\) such a value that \(n\sqrt{y} - 1\) is a fraction less than unity, we render both the series for the value of log \(y\) converging; for then the expression \(1 - \frac{1}{n\sqrt{y}}\) will also be less than unity, seeing it is equal \(\frac{n\sqrt{y} - 1}{n\sqrt{y}}\). Therefore, in the first series, the second and third terms (taken together as one term) constitute 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 log \(y\). Hence

\[\log y < nM\left(n\sqrt{y} - 1\right).\]

And since, on the contrary, the terms of the second series 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 log \(y\); so that

\[\log y > nM\left(1 - \frac{1}{n\sqrt{y}}\right).\]

Thus we have two limits to the value of the logarithm of \(y\), which, by taking the number \(n\) sufficiently great 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.

\[\frac{1}{M} < n\left(n\sqrt{10} - 1\right), \quad \frac{1}{M} > n\left(1 - \frac{1}{n\sqrt{10}}\right).\]

It is evident that the difference between the two limits of log \(y\), is

\[nM\left\{(n\sqrt{y} - 1) - \left(1 - \frac{1}{n\sqrt{y}}\right)\right\},\]

therefore; if we take either the one or the other of the two preceding expressions for log \(y\), the error in excess or defect is necessarily less than this quantity.

By these formulas we may depend upon having the logarithm of any number true to \(m\) figures, if we give to \(n\) such a value that the root \(n\sqrt{y}\) shall have \(m\) cipher between the decimal point and the first significant figure on the right. So that in general, as the error is the smaller as 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,

\[nM\left(n\sqrt{y} - 1\right), \quad nM\left(1 - \frac{1}{n\sqrt{y}}\right)\]

is the accurate value of 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 \(n\sqrt{y}\) may be found by a repetition of extractions 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 cipher 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

\[\frac{1}{n} \times \frac{1}{n\sqrt{10} - 1}\]

nearly,

when \(n\) is some very great number. Let us suppose \(n = 2^{20} = 8^{10}\); then, dividing unity by 8, and this result again by 8, and so on, we shall, after 20 divisions, have \(\frac{1}{n}\) or \(\frac{1}{8^{10}}\) equal to

0-00000 00000 00000 00086 73817 37988 40354.

Also, by extracting the square root of 10, and the square root of this result, and so on, after performing 60 extractions we shall find \(\sqrt[60]{10}\) equal to

1-00000 00000 00000 00000 00159 71742 08125 50527 03251.

Therefore, \(\frac{1}{n} \times \frac{1}{n\sqrt{10} - 1}\) or \(M\) is equal to

86796173798840354 = 0-4342944819.

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

\[nM\left(n\sqrt{3} - 1\right) = n\left(n\sqrt{\frac{3}{10} - 1}\right) = n\sqrt{\frac{3}{10} - 1}.\]

\(n\) being as before a very great number. Let us suppose also in this case that \(n = 2^{20}\); then after 60 extractions of the square root we have \(n\sqrt{3}\) equal to

1-00000 00000 00000 0095 28942 6074 58932.

VOL XXXIII. Therefore, taking the value of \( \sqrt[10]{10} \) as found in last example, we have

\[ \log_3 = \frac{n^3 - 1}{\sqrt{10} - 1} = \frac{95289426407458982}{199717420812550527} = 477121254719662. \]

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 \( \sqrt[10]{y} \) a value which is unity followed by a decimal fraction; and then \( \sqrt[10]{y} - 1 \) 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 series by which the series

\[ \log_y = M \left( y - 1 - \frac{1}{2}(y-1)^2 + \frac{1}{3}(y-1)^3 - \frac{1}{4}(y-1)^4 + \cdots \right) \]

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

\[ \log_y (1+u) = M \left( u - \frac{u^2}{2} + \frac{u^3}{3} - \frac{u^4}{4} + \frac{u^5}{5} - \cdots \right). \]

In like manner, if \( 1-u \) be substituted for \( y \), we have

\[ \log_y (1-u) = M \left( -u - \frac{u^2}{2} - \frac{u^3}{3} - \frac{u^4}{4} - \frac{u^5}{5} - \cdots \right). \]

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

\[ \log_y \frac{1+u}{1-u} = 2M \left( u + \frac{u^3}{3} + \frac{u^5}{5} + \frac{u^7}{7} + \cdots \right), \]

which series, by substituting \( z = \frac{1+u}{1-u} \) and consequently \( z = \frac{1}{z+1} \) for \( u \), will be otherwise expressed thus,

\[ \log_y z = 2M \left( z - \frac{1}{z+1} + \frac{1}{2} \left( \frac{1}{z+1} \right)^2 + \frac{1}{3} \left( \frac{1}{z+1} \right)^3 + \cdots \right); \]

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

\[ 2 \left( \frac{1}{3} + \frac{1}{3^3} + \frac{1}{3^5} + \frac{1}{3^7} + \cdots \right) = A + \frac{1}{2}B + \frac{1}{2}C + \frac{1}{2}D + \frac{1}{2}E + \cdots \]

where \( A \) is put for \( \frac{2}{3} \), \( B \) for \( \frac{2}{3^3} \), \( C \) for \( \frac{2}{3^5} \), \( D \) for \( \frac{2}{3^7} \), etc. The calculation will be as follows:

\[ A = 0.666666666666 \\ B = 0.074074074074 \\ C = 0.008230452304 \\ D = 0.000914494742 \\ E = 0.000101610161 \\ F = 0.000011290059 \\ G = 0.000001254545 \\ H = 0.000000189383 \\ I = 0.000000015487 \\ K = 0.000000001721 \\ L = 0.0000000000191 \\ M = 0.0000000000021 \]

Nap. log. 2 = 0.693147180553

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 13th place, being 0.693147180553.

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, etc. In computing the logarithm of 3, the series would converge by the powers of the fraction \( \frac{3-1}{3+1} = \frac{1}{2} \); and in computing the logarithm of 5 it would converge by the powers of \( \frac{5-1}{5+1} = \frac{2}{3} \); 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 \( \frac{1+u}{1-u} = 1 + \frac{z}{n} \), and consequently \( u = \frac{z}{2n+z} \); these values being substituted in the formula, \( \log_y \frac{1+u}{1-u} = 2M \left( u + \frac{u^3}{3} + \frac{u^5}{5} + \frac{u^7}{7} + \cdots \right) \)

we have \( \log_y \left( 1 + \frac{z}{n} \right) \) equal to

\[ 2M \left( \frac{z}{2n+z} + \frac{1}{2} \left( \frac{z}{2n+z} \right)^2 + \frac{1}{3} \left( \frac{z}{2n+z} \right)^3 + \cdots \right). \]

but \( \log_y \left( 1 + \frac{z}{n} \right) = \log_y \frac{n+z}{n} = \log_y (n+z) - \log_y n \),

therefore, \( \log_y (n+z) = \log_y n + 2M \left( \frac{z}{2n+z} + \frac{1}{2} \left( \frac{z}{2n+z} \right)^2 + \frac{1}{3} \left( \frac{z}{2n+z} \right)^3 + \cdots \right) \).

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 \( \frac{z}{2n+z} = \frac{1}{3} \).

Therefore the logarithm of 3 is

\[ \log_2 3 = 2 \left( \frac{1}{3} + \frac{1}{3^3} + \frac{1}{3^5} + \frac{1}{3^7} + \cdots \right) = \log_2 2 + A + \frac{1}{2}B + \frac{1}{2}C + \frac{1}{2}D + \frac{1}{2}E + \cdots \] The calculation may stand thus:

\[ \begin{align*} A &= 400000000000 \\ B &= 053333333333 \\ C &= 001280000000 \\ D &= 000003657143 \\ E &= 000000113778 \\ F &= 000000003724 \\ G &= 000000000136 \\ H &= 000000000004 \\ \end{align*} \]

Nap. log. \(2 = 0.693147180551\) Nap. log. \(3 = 1.098612288659\) Nap. log. \(6 = 1.791759489210\)

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 \(\frac{2 \times 5^5}{7^2} = \frac{50}{49}\), therefore \(\log_2 + 2 \log_5 - 2 \log_7 = \log_2 \frac{50}{49}\), and consequently

\[ \log_7 = \frac{1}{2} \log_2 + \log_5 - \frac{1}{2} \log_2 \frac{50}{49} \]

Now the logarithm of \(\frac{50}{49}\) may be readily obtained from

\[ \log_2 = 2M \left\{ \frac{z-1}{z+1} + \frac{1}{2} \left( \frac{z-1}{z+1} \right)^2 + \frac{1}{3} \left( \frac{z-1}{z+1} \right)^3 + \ldots \right\} \]

For, substituting \(\frac{50}{49}\) for \(z\), the formula gives

\[ \text{Nap. log. } \frac{50}{49} = 2 \left( \frac{1}{49} + \frac{1}{3 \cdot 49^2} + \frac{1}{5 \cdot 49^3} + \ldots \right) \]

where \(A = \frac{1}{9 \cdot 11}\), \(B = \frac{A}{9 \cdot 11^2}\), \(C = \frac{B}{9 \cdot 11^3}\), etc. 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.

\[ \begin{align*} A &= 0.020202020202 \\ B &= \frac{1}{9 \cdot 11} A = 0.00002061220 \\ C &= \frac{1}{9 \cdot 11^2} B = 0.00000000210 \\ A &= 0.020202020202 \\ \frac{1}{2} B &= 0.00000087073 \\ \frac{1}{3} C &= 0.00000000042 \\ \end{align*} \]

Nap. log. \(50/49 = 0.020202707317\)

\[ \frac{1}{2} \log_2 = 0.346573590275 \\ \log_5 = 1.609437912417 \\ \]

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:

Nap. log. \(2 = 0.693147180551\) Nap. log. \(3 = 1.098612288659\) Nap. log. \(9 = 2.197224577318\) Nap. log. \(2 = 0.693147180551\) Nap. log. \(5 = 1.609437912417\) Nap. log. \(10 = 2.302585092968\) 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

\[ \frac{1}{2302585092968} = 4342944819. \]

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.

\[ M = 434294481903251827651128918916 \] \[ \frac{1}{M} = 2302585092994045684017991454684 \]

The formulae 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.

Because

\[ \log_e z = 2M \left\{ \frac{z-1}{z+1} + \frac{1}{2} \left( \frac{z-1}{z+1} \right)^2 + \frac{1}{3} \left( \frac{z-1}{z+1} \right)^3 + &c. \right\} \]

If we now suppose

\[ z = \frac{n^2}{n^2 - 1} = (n-1)(n+1), \]

so that \( \frac{z-1}{z+1} = \frac{1}{2n^2-1} \), then the formula becomes

\[ \log_e \frac{n^2}{(n-1)(n+1)} = 2M \left\{ \frac{1}{2n^2-1} + \frac{1}{3} \left( \frac{1}{2n^2-1} \right)^2 + &c. \right\} \]

But \( \log_e \frac{n^2}{(n-1)(n+1)} = 2 \log_e n - \log_e (n-1) - \log_e (n+1) \), therefore, putting N for the series

\[ 2M \left\{ \frac{1}{2n^2-1} + \frac{1}{3} \left( \frac{1}{2n^2-1} \right)^2 + &c. \right\} \]

we have this formula,

\[ 2 \log_e n - \log_e (n-1) - \log_e (n+1) = N; \]

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 \( \log_9 9 = 0.95424250943 \), \( \log_{10} 10 = 1 \); to find the common logarithm of 11.

Here we have \( n = 10 \), so that the formula gives in this case \( 2 \log_{10} 10 - \log_9 9 - \log_{10} 11 = N \), and hence we have

\[ \log_{10} 11 = 2 \log_{10} 10 - \log_9 9 - N, \]

where \( N = \frac{2M}{199} + \frac{2M}{3 \cdot 199} + &c. \)

\( M \) being 43429448190.

Calculation of N.

\[ A = \frac{2M}{199} = 0.00436476856 \] \[ B = \frac{A}{3 \cdot 199} = 0.00000003674 \] \[ N = 0.00436480540 \] \[ \log_9 9 = 0.95424250943 \] \[ \log_9 9 + N = 0.95860731483 \] \[ 2 \log_{10} 10 = 2.00000000000 \] \[ \log_{10} 11 = 1.04139268517 \]

Here the series expressed by N converges very fast, so that two of its terms are sufficient to give the logarithm of 11 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 × 7, 99 = 9 × 11, and 100 = 10 × 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

\[ 2 \log_9 99 - \log_9 98 - \log_9 100 = N, \]

that is, substituting log_9 9 + log_9 11 for log_9 99, log_9 2 + 2 log_7 for log_9 98, and 2 log_10 10 for log_9 100,

\[ 2 \log_9 9 + 2 \log_9 11 - \log_9 2 - 2 \log_7 - 2 \log_10 = N, \]

and hence by transposition, &c.

\[ \log_9 11 = \frac{1}{2} N + \frac{1}{2} \log_9 2 + \log_9 7 - \log_9 9 + \log_9 10; \]

and in this equation

\[ N = \frac{2M}{19901} + \frac{2M}{3 \cdot 19901} + &c. \]

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

\[ \log_e z = 2M \left\{ \frac{z-1}{z+1} + \frac{1}{2} \left( \frac{z-1}{z+1} \right)^2 + \frac{1}{3} \left( \frac{z-1}{z+1} \right)^3 + &c. \right\} \]

put \( z = \frac{(n-1)^2(n+2)}{(n-2)(n+1)^2} = \frac{n^2-3n+2}{n^2-3n-2} \),

then \( \frac{z-1}{z+1} = \frac{2}{n^2-3n} \).

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

\[ \log_e \frac{(n-1)^2(n+2)}{(n-2)(n+1)^2} = 2M \left\{ \frac{2}{n^2-3n} + \frac{1}{3} \left( \frac{2}{n^2-3n} \right)^2 + &c. \right\} \]

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

\[ 2M \left\{ \frac{2}{n^2-3n} + \frac{1}{3} \left( \frac{2}{n^2-3n} \right)^2 + &c. \right\} \]

we have this formula,

\[ \log_e (n+2) + 2 \log_e (n-1) - \log_e (n-2) - 2 \log_e (n+1) = P. \]

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 \( \log_6 6 = \log_2 2 + \log_3 3 \), and \( \log_8 8 = 3 \log_2 2 \), we have these four equations,

\[ \log_7 7 + 2 \log_2 2 - 3 \log_3 3 = \frac{2M}{55} + \frac{2M}{3 \cdot 55} + &c. \] \[ -2 \log_7 7 + \log_2 2 + 2 \log_5 5 = \frac{2M}{99} + \frac{2M}{3 \cdot 99} + &c. \] \[ 4 \log_3 3 - 4 \log_2 2 - \log_5 5 = \frac{2M}{161} + \frac{2M}{3 \cdot 161} + &c. \] \[ \log_5 5 - 5 \log_3 3 + 2 \log_7 7 = \frac{2M}{244} + \frac{2M}{3 \cdot 244} + &c. \]

Let \( \log_2 2, \log_3 3, \log_5 5, \) and \( \log_7 7 \), be now considered as Construct four unknown quantities, and by resolving those equations in the usual manner, the logarithms may be determined.

Resuming once more the formula

\[ \log z = 2M \left\{ \frac{z-1}{z+1} + \frac{1}{3} \left( \frac{z-1}{z+1} \right)^3 + \text{etc.} \right\} \]

let \( n(n+5)(n-5) \) be substituted in it instead of \( z \), then, by this substitution, \( \frac{z-1}{z+1} \) will become

\[ \frac{-72}{n^4 - 25m^4 + 72} \] and the formula will be transformed to

\[ \log \frac{n(n+5)(n-5)}{(n+3)(n-3)(n+4)(n-4)} = -2M \left\{ \frac{72}{n^4 - 25m^4 + 72} + \frac{1}{3} \left( \frac{72}{n^4 - 25m^4 + 72} \right)^3 + \text{etc.} \right\} \]

Hence, putting the latter side of this equation equal to \( Q \), we have this formula,

\[ 2 \log n + \log (n+5) + \log (n-5) - \log (n+3) - \log (n-3) - \log (n+4) - \log (n-4) + Q = 0 \]

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,

\[ \log 1231 = \log (1230 + 1) = \log \left\{ 1230 \left( 1 + \frac{1}{1230} \right) \right\} \]

\[ = \log 1230 + \log \left( 1 + \frac{1}{1230} \right) \]

Again, \( \log 1230 = \log 2 + \log 5 + \log 123 \), and \( \log 123 = \log \left\{ 120 \left( 1 + \frac{1}{40} \right) \right\} = \log 120 + \log \left( 1 + \frac{1}{40} \right) \).

\[ \log 120 = \log (2^3 \times 3 \times 5) = 3 \log 2 + \log 3 + \log 5 \]

Therefore \( \log 1231 = 4 \log 2 + 2 \log 3 + 2 \log 5 + \log \left( 1 + \frac{1}{40} \right) + \log \left( 1 + \frac{1}{1230} \right) \]

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

\[ 1 + \frac{1}{40}, 1 + \frac{1}{1230} \]

all of which may be easily found by the formulae already delivered.

### III.—NATURE AND CALCULATION OF SINES, TANGENTS, &c.

The trigonometrical formulae given in the Arithmetic of Sines (See Algebra) contain the principles requisite for constructing tables in which the lengths of the sines, cosines, tangents, cotangents, secants, and cosecants of given angles, are exhibited for every degree, minute, and second of space of the quadrant. Such tables are known as Trigonometrical Canons. It is evident that such tables, when accurately computed, are invaluable, and will prevent the labour and trouble of calculating afresh, on every new occasion, the sine, cosine, &c., of any angle that may be required.

Trigonometrical Canons are generally calculated to a radius unity, or 1; but when this is the case, all the results are multiplied by 10,000, or any high power of 10, in order to avoid printing cyphers to the immediate right of the decimal point of the sines of all small angles.

Since the length of the semi-circumference of a circle has been approximated to, and found to be \( \pi = 3.1415926535897932 \) when radius = 1; and since the number of degrees in the semi-perimeter is 180° = 10800; hence 1' of space = 1' of an arc = 0.000290888204. Now, since the small angle of 1' is subtended by an arc of the same magnitude, the sine of 1' will also be a small quantity, and may be regarded as equal to the arc. Thus the sine 1' = length of an arc of 1' = 0.000290888204. In this manner the sine of 1' may be determined, but more legitimately from the formula of \( \cos 2A = 2 \cos^2 A - 1 \).

\[ \therefore \cos A = \sqrt{\frac{1}{2}(1 + \cos 2A)}, \text{when } A \text{ not } > 90^\circ; \text{ and since this is true whatever the angle } A \text{ be, it is true when for } A \text{ we write } \frac{A}{2}. \text{ Hence,} \]

\[ \cos \frac{A}{2} = \sqrt{\frac{1}{2}(1 + \cos A)}; \quad \cos \frac{A}{2} = \sqrt{\frac{1}{2}(1 + \cos \frac{A}{2})}; \]

\[ \cos \frac{A}{2} = \sqrt{\frac{1}{2}(1 + \cos \frac{A}{2})}; \quad \cos \frac{A}{2} = \sqrt{\frac{1}{2}(1 + \cos \frac{A}{2})}; \]

\[ \text{etc.} = \text{etc.} = \text{etc.} = \text{etc.} \]

Let \( A \) receive a particular value, as 30°; then \( \cos 2A = \cos 60° = \frac{1}{2} \). Therefore,

\[ \cos 30° = \sqrt{\frac{1}{2}(1 + \cos 60°)} = \sqrt{\frac{1}{2}(1 + \frac{1}{2})} = \frac{\sqrt{3}}{2} \]

\[ = 0.8660254; \]

\[ \cos 15° = \sqrt{\frac{1}{2}(1 + \cos 30°)} = \sqrt{\frac{1}{2}(1 + 0.8660254)} \]

\[ = 0.9659258; \]

\[ \cos 7° 30' = \sqrt{\frac{1}{2}(1 + \cos 15°)} = \sqrt{\frac{1}{2}(1 + 0.9659258)} \]

\[ = 0.9914449; \]

\[ \text{etc.} = \text{etc.} = \text{etc.} \]

If now we proceed in this manner and compute eleven bisections of the angle 30°; that is, \( \frac{30°}{2^n} \), we shall at last come to \( \cos \frac{30°}{2^{11}} = \sqrt{\frac{1}{2}(1 + \cos \frac{30°}{2^{11}})} \), or

\[ \cos \frac{30°}{2^{11}} = \cos \frac{295}{256} \times 1' = \cos 52°734375; \]

\[ \therefore \cos 52°734375 = 0.99999996782, \text{ which is a very little greater than the cosine of } 1'; \]

\[ \therefore \sin \frac{30°}{2^{11}} = \sin 52°734375 = \sqrt{1 - \cos^2 \frac{30°}{2^{11}}} = 0.000255663462. \]

But the sines of very small angles increasing and decreasing as the angles themselves, that is, as the number of minutes they contain,

\[ \therefore \sin 1' : \sin \frac{30°}{2^{11}} = 1' : \frac{30°}{2^{11}} = \frac{30°}{2^{11}} \times \frac{60 \times 1'}{256} \]

\[ = 1': \frac{225}{256} = 1: 0.000290888204 \]

\[ \therefore \sin 1' = 0.000290888204. \]

So also \( \cos 1' = \sqrt{1 - \sin^2 1'} = \sqrt{1 - (0.000290888204)^2} = 0.9999999577 \).

If \( \sin 30° \) be determined from the formula,

\[ \sin A = \frac{1}{2} \left\{ \sqrt{1 + \sin 2A} - \sqrt{1 - \sin 2A} \right\}, \text{we have} \]

\[ \sin 30° = \frac{1}{2} \left\{ \sqrt{1 + \sin 1'} - \sqrt{1 - \sin 1'} \right\} \]

\[ = 0.000145199, \text{ which is very nearly equal to half the sine of } 1', \text{ since the angle is small}; \]

\[ \therefore \cos 30° = \sqrt{1 - \cos^2 30°} = \sqrt{1 - (0.000145199)^2} = 0.9999999996. \]

Since the sine and cosine of 1' and 30° are known, the sines and cosines of all angles from 1' or 30° up to 90°, differing from one another by 1' or 30°, can easily be determined. This is effected by the convenient formulae of

\[ \sin nA = 2 \sin (n-1)A \cdot \cos A - \sin (n-2)A, \text{ and} \]

\[ \cos nA = 2 \cos (n-1)A \cdot \cos A - \cos (n-2)A, \]

in which, make \( A = 1' \), and for \( n \) write successively the natural numbers 2, 3, &c., and Now it would be very inconvenient to register so many cyphers which necessarily enter the values of the sines of small angles; therefore, in order to avoid this, the sines, cosines, and the quantities which are derived from them, are multiplied by 10,000, or a suitable power of 10; that is, the decimal point is removed four or more places to the right, previous to their being registered in the tables. If the real values of the sines, cosines, and the other quantities dependent thereon, be tabulated, without being multiplied by any power of 10, the tables are known as Tables of Natural Sines, Cosines, Tangents, &c.

In this manner, therefore, we can easily derive from the preceding results the sines and cosines of all angles from 1° to 90°.

The same quantities may also be obtained from the formula \( \sin(A + B) = \sin A + \{ \sin A - \sin(A - B) \} - \sin A \left( 2 \sin \frac{B}{2} \right)^2 \); so that if \( B = 1° \), and if for \( A \) we write successively 1°, 2°, 3°, &c., we have, e.g., when \( A = 1° \), and \( B = 1° \),

\[ \sin 2° = \sin 1° + (\sin 1° - \sin 0°) - \sin 1° (2 \sin 30°)^2. \]

In either case, we shall have a long multiplication to perform, but the same result.

But we need not proceed in these calculations beyond the sine and cosine of 45°, for the same values will recur from 45° to 90°. The reason of this is because \( \sin(45° + A) = \cos(90° - (45° + A)) = \cos(45° - A) \); and also

\[ \cos(45° + A) = \sin(90° - (45° + A)) = \sin(45° - A). \]

Hence a trigonometrical canon of sines and cosines is completed at 45°.

When, however, the sines and cosines have been computed from 1° up to 30°, the labour and risk of error in finding long products may be much diminished, by determining the value of the sine or cosine by addition and subtraction. Thus, we have

\[ 2 \sin A \cdot \cos B = \sin(A + B) + \sin(A - B), \]

and

\[ 2 \sin A \cdot \sin B = \cos(A - B) - \cos(A + B). \]

Let \( A = 30° \), then \( 2 \sin A = 2 \sin 30° = 1 \);

\[ \therefore \cos B = \sin(30° + B) + \sin(30° - B) \quad (\text{i}), \]

and

\[ \sin B = \cos(30° - B) - \cos(30° + B) \quad (\text{ii}). \]

In these expressions, let \( B \) increase by 1° at a time, from 0° up to 30°, and we shall have all sines and cosines from 30° up to 60°; for

\[ \begin{align*} \sin 30° 1' &= \cos 1° - \sin 29° 59' \\ \sin 30° 2' &= \cos 2° - \sin 29° 58' \\ &\vdots \\ \cos 30° 1' &= \cos 29° 59' - \sin 1° \\ \cos 30° 2' &= \cos 29° 58' - \sin 2° \\ &\vdots \end{align*} \]

Again, in order to derive the sines and cosines of all angles from 60° up to 90°, we have

\[ 2 \cos A \cdot \sin B = \sin(A + B) - \sin(A - B), \]

and

\[ 2 \cos A \cdot \cos B = \cos(A + B) + \cos(A - B). \]

Let \( A = 60° \), then \( 2 \cos A = 2 \cos 60° = 1 \);

\[ \therefore \sin B = \sin(60° + B) - \sin(60° - B) \quad (\text{iii}), \]

and

\[ \cos B = \cos(60° + B) + \cos(60° - B) \quad (\text{iv}). \]

Give \( B \) all values from 0° up to 30°, the increase being by 1° at a time;

\[ \sin 60° 1' = \sin 1° + \sin 59° 59' \]

When the sines and cosines of all angles from 0° up to 90° have been computed, they form the basis for calculating the tangents and cotangents, the secants and cosecants, of all angles from 0° up to 90°. For \( \tan A = \frac{\sin A}{\cos A} \); \( \cot A = \frac{\cos A}{\sin A} \); \( \sec A = \frac{1}{\cos A} \); \( \csc A = \frac{1}{\sin A} \).

But if we suppose that the tangents of all angles from 0° up to 45° have been already tabulated, those from 45° up to 90° may be obtained very simply from the formula

\[ 2 \tan 2A = \tan(45° + A) - \tan(45° - A). \]

Let now \( A \) receive increments of 1° at a time;

\[ \tan 45° 1' = 2 \tan 2° + \tan 44° 59' \]

\[ \tan 45° 2' = 2 \tan 4° + \tan 44° 58' \]

\[ \tan 45° 3' = 2 \tan 6° + \tan 44° 57' \]

&c.

The tangents of all angles being known, the cotangents are deduced therefrom; and although the sines and cosines will enable us to find the cosecants and secants respectively, yet the same quantities will be found much more simply by the following formulae:

\[ 2 \sec A = \tan \left( 45° + \frac{A}{2} \right) + \cot \left( 45° + \frac{A}{2} \right), \quad \text{and} \]

\[ 2 \csc A = \tan \frac{A}{2} + \cot \frac{A}{2}. \]

Let \( A \) increase by 1° at a time, then, on sec \( A \) and cosec \( A \) becoming sec 45°, and cosec 45°, we shall have the one the complement of the other, since sec \( A = \csc(90° - A) \); hence,

\[ \begin{align*} \sec 1° &= \frac{1}{2} (\tan 45° 0' 30'' + \cot 45° 0' 30'') \\ \sec 2° &= \frac{1}{2} (\tan 45° 1' + \cot 45° 1') \\ \sec 3° &= \frac{1}{2} (\tan 45° 1' 30'' + \cot 45° 1' 30'') \\ &\vdots \\ \csc 1° &= \frac{1}{2} (\tan 30° + \cot 30°) \\ \csc 2° &= \frac{1}{2} (\tan 1° + \cot 1°) \\ \csc 3° &= \frac{1}{2} (\tan 1° 30'' + \cot 1° 30'') \\ &\vdots \end{align*} \]

We have already seen, that with respect to the sines and cosines, the tabulated results need be carried no further than 45°; the same may be said of the other elements which have been mentioned. For \( \tan A = \cot(90° - A) \); \( \sec A = \csc(90° - A) \). Hence, at the top of trigonometrical tables the angles increase forwards from 0° to 45°, while at the bottom they proceed backwards from 45° to 90°.

In the construction of these trigonometrical canons, however, it is absolutely necessary to have formulae of verification, or checks to verify, or rather to examine, the accuracy of the results as they progress from 0° up to 45°; for an error in one result is communicated to each quantity. These formulae of verification are, besides i. ii. iii. iv.—

\[ \begin{align*} \cos(35° + A) &= \cos(25° - A) + \cos(35° - A) - \cos(75° + A) = \cos A; \\ \sin A + \sin(35° - A) + \sin(75° + A) &= \sin(35° + A) + \sin(75° - A); \\ \sin(55° + A) + \sin(35° - A) - \sin(18° + A) - \sin(18° - A) &= \cos A; \\ &\vdots \end{align*} \]

Suppose now that the sines, cosines, &c., of all angles have been calculated, radius being unity, and that we multiply each natural sine, cosine, &c., by 10,000,000,000 = 10^9, the logarithm of which is 10, then if of each of these new values we take the logarithm, and register such values in tables, we shall have a logarithmic canon of sines, cosines, &c.; and, therefore, the logarithms of the sines instead of the sines themselves. These logarithmic sines, &c., are found to be much more convenient in calculation than the natural sines, which are hence omitted in logarithmic tables. The real logarithm is found by subtracting 10 from the characteristics of the tabulated logarithms.