Fig. 5. Fig. 6. Fig. 7. Fig. 8. Fig. 9. Fig. 10. Fig. 11.

E. Mitchell fecit. North Pole, p. 97, of two other logs, which were tried by Captain Phipps: one invented by Mr Ruffel, the other by Foxon; both constructed upon this principle, that a spiral, in proceeding its own length in the direction of its axis through a resisting medium, makes one revolution round the axis; if therefore the revolutions of the spiral are registered, the number of times it has gone its own length through the water will be known. In both these the motion of the spiral in the water is communicated to the clockwork within board, by means of a small line fastened at one end to the spiral, which tows it after the ship, and at the other to a spindle, which sets the clockwork in motion. That invented by Mr Ruffel has a half spiral of two threads, made of copper, and a small dial of clockwork, to register the number of turns of the spiral. The other log has a whole spiral of wood with one thread, and a larger piece of clockwork with three dials, two of them to mark the distance, and the other divided into knots and fathoms, to show the rate by the half-minute glass, for the convenience of comparing it with the log. This kind of log will have the advantage of every other in smooth water and moderate weather; and it will be useful in finding the trim of a ship when alone, in surveying a coast in a single ship, or in measuring distances in a boat between headlands and shoals; but it is subject to other inconveniences, which will not render it a proper substitute for the common log.

Perpetual Log, a machine so called by its inventor, Mr Gottlieb of London, is intended for keeping a constant and regular account of the rate of a ship's velocity in the interval of heaving the log.

Fig. 1. is a representation of the whole machine; the lower part of which, EFG, is fixed to the side of the keel; H representing only the boundary line of the ship's figure. EF are the section of a wooden external case, left open at the ends KL, to admit the passage of the water during the motion of the ship. At M is a copper grating, placed to obstruct the entrance of any dirt, &c. into the machine. I is a section of a water wheel, made from 6 to 12 inches in diameter, as may be necessary, with floatboards upon its circumference, like a common water wheel, that turn by the resistance of the water passing through the channel LK. It turns upon a shouldered axis, represented by the vertical section at K. When the ship is in motion, the resistance of the water through the channel LK turns round the wheel I. This wheel, by means of a pinion, is connected with and turns the rod contained in the long copper tube N. This rod, by a pinion fixed at its upper extremity, is connected with and turns upon the whole system of wheels contained in the dial of the case ABCD. This dial, by means of the copper tube N, may be fixed to any convenient place aboard the ship. In the front of the dial are several useful circular graduations, as follow: The reference by the dotted line A has a hand which is moved by the wheels within, which points out the motion of the ship in fathoms of 6 feet each. The circle at B has a hand showing the knots, at the rate of 48 feet for each knot: and is to be observed with the half-minute glass at any time. The circle at C has a short and a long hand; the former of which points out the mile in land measure, and the latter or longer the number of knots contained in each mile, viz. 128, which is in the same proportion to a mile as 60 minutes to the hour in the reckoning. At e, a small portion of a circle is seen through the front plate called the register; which shows, in the course of 24 hours (if the ship is upon one tack) the distance in miles that the ship has run; and in the 24 hours the mariner need take but one observation, as this register serves as an useful check upon the fathoms, knots, and miles, shewn upon the two other circles.

f Is a plate showing 100 degrees or 6000 miles, and also acts as another register or check; and is useful in case of any mistake being made in observing the distance run by the other circles. The reckoning by these circles, without fear of mistake, may therefore be continued to nearly 12,000 miles.

A communication from this machine may easily be made to the captain's bedfide, where by touching a spring only, a bell in the head ABCD will sound as many times in a half minute as the ship sails miles in an hour.

Log-Board, a sort of table, divided into several columns, containing the hours of the day and night, the direction of the winds, the course of the ship, and all the material occurrences that happen during the 24 hours, or from noon to noon; together with the latitude by observation. From this table the officers of the ship are furnished with materials to compile their journals.

Log-Book, a book into which the contents of the log-board is daily copied at noon, together with every circumstance deserving notice that may happen to the ship, either at sea or in a harbour. See Navigation.

INTRODUCTION.

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 16th century, and the beginning of the 17th, 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, which were so adapted to the numbers to be multiplied, or divided, that these being arranged in the form of a table, each opposite to the number number called its logarithm, the product of any two numbers in the table was found by the addition of their logarithms; and, on the contrary, the quotient arising from the division of one number by another was found 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, the first term of which is unity or 1, and the common ratio any number whatever, and the other an arithmetical progression, the first term of which is 0, and the common difference also any number whatever; (but as a particular example we shall suppose the common ratio of the geometrical series to be 2, and the common difference of the arithmetical series 1), and let the two series be written opposite to each other in the form of a table, thus:

<table> <tr> <th>Geom. Prog.</th> <th>Arith. Prog.</th> </tr> <tr> <td>1</td> <td>0</td> </tr> <tr> <td>2</td> <td>1</td> </tr> <tr> <td>4</td> <td>2</td> </tr> <tr> <td>8</td> <td>3</td> </tr> <tr> <td>16</td> <td>4</td> </tr> <tr> <td>32</td> <td>5</td> </tr> <tr> <td>64</td> <td>6</td> </tr> <tr> <td>128</td> <td>7</td> </tr> <tr> <td>256</td> <td>8</td> </tr> <tr> <td>512</td> <td>9</td> </tr> <tr> <td>1024</td> <td>10</td> </tr> <tr> <td>2048</td> <td>11</td> </tr> <tr> <td>4096</td> <td>12</td> </tr> <tr> <td>&amp;c.</td> <td>&amp;c.</td> </tr> </table>

The two series being thus arranged, 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 equal to 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 terms of the arithmetical series corresponding to them (that is, their logarithms) are 2 and 5; now the product of the numbers is 1:8, and the sum of their logarithms is 7; and it appears by inspection of the two series, that the latter number is the logarithm of the former, agreeing with the proposition we are illustrating. In like manner, if the numbers or terms of the geometrical series be 16 and 64, the logarithms of which are 4 and 6, we find from the table that \(10-4+6\) is the logarithm of \(1024=16\times64\); 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 equal to the logarithm of that term of the series which is equal to 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 \(1024\div16=64\), and that \(10-4=6\); now it appears from the table that 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 the logarithm of 16, which is 4; we add the logarithm of 128, which is 7, and opposite to the sum 11, we find 2048, the product sought. On the other hand, to divide any number in the table by any other number, we must subtract the logarithm of the divisor from that of the dividend, and look for the remainder among the logarithms, and opposite to it we shall find the number sought. Thus, to divide 2048 by 128; from 11, which is the logarithm of 2048 we subtract 7, the logarithm of 128, and opposite to 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 the logarithms 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, among 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 of the geometrical progression were found together with their logarithms, as were either accurately equal to, or coincided nearly with, all numbers bers within certain limits (for example between 1 and 100000), 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 Jusfe Byrge, two mathematicians who were cotemporary with Lord Napier; but there is no reason to suppose that either of these anticipated him, for Longomontanus never published any thing on the subject, although he lived thirty-three years after Napier had made known his discovery; and as to Byrge, he is indeed known to have printed a table containing an arithmetical and a geometrical progression written opposite to each other, so as to form in effect a system of logarithms of the same kind as those invented by Napier, without however explaining their nature and use, although it appears from the title he intended to do so, but was probably prevented by some cause unknown to us. But this work was not printed till 1620, fix years after Napier had published his discovery.

It is therefore with good reason that Napier is now universally considered as the first, and most probably as the only inventor. This discovery he published in the year 1614 in a book entitled Mirifici Logarithmorum Canonis Descriptio, but he referred the construction of the numbers till the opinion of the learned concerning his invention should be known. 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 fine, 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 becomes slower and slower, and such, 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 shew that at the end of any successive equal intervals of time from the beginning of the motion, the portions of the line which remain will constitute a decreasing geometrical progression.

Again, he supposes another line to be generated by a point which moves along it equally, or which passes over equal intervals of it in equal times. Thus the portions of the line generated at the end of any equal successive intervals of time from the beginning of the motion will form a series of quantities in arithmetical progression. Now if the two motions be supposed to begin together, at the end of any equal intervals of time the remainders of the one line will form a series of quantities in geometrical progression, and 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 infinite.

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 shall have different systems of logarithms. Napier supposed the 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 disused, and a more convenient one substituted instead of it, 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 the inventor of the principles of what is commonly though erroneously called Mercator's sailing. The translation was sent to Napier for his perusal, and returned with his approbation, and 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 the year 1616, accompanied with a dedication by his son to the East India Company, and a preface by Henry Briggs, who afterwards distinguished himself so much by his improvement 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 the logarithms of numbers, and the contrary, 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 the year 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 the year 1624, and which contained the logarithms of the whole numbers 1, 2, 3, 4, &c. to 1000, together with their differences and arithmetical complements, &c. This table is now 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, (see Fluxions, § 152. Ex. 5.).

In 1719 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 Confructio, and other pieces written by his father and Mr Briggs. An exact copy of the fame two works in one volume was also printed in 1625 at Lyons in France. In 1618 or 1619 Benjamin Ursinus, mathematician to the elector of Brandenburg, published Napier's tables of logarithms in his Curius Mathematicus, to which he added some tables of proportional parts; and in 1624, he printed his Trigonometria, with a table of natural fines, 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 Demonstratio legitima Ortus Logarithmorum eorumque Uso, &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 of them who were somewhat advanced in years, and were grown averse to new methods of reasoning that carried them out of the old doctrines and principles with which habit had rendered them familiar, doubted in some degree 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 considered Napier's demonstration of it as illegitimate and unsatisfactory. 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 which the German mathematicians had objected to (and not without reason) in Napier's mode of treating the subject.

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 (whom we have already mentioned) applied himself with great earnestness to their study and improvement, and it appears that he had projected at an early period that advantageous change in the system which has since taken place. 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, (2, 1, 10, 100, 1000, &c.) formed a decreasing arithmetical series, the common difference of the terms of which was 2.3025851. 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 see him, and to converse with him upon 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 fines to increase from 0 (the logarithm of radius) to infinity, while the fines 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 100000, &c. should be the logarithm of radius; and 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 Briggs was the inventor of this improved system of logarithms which has since been universally adopted, and that the only share that Napier had in it was his suggestion to Briggs to begin with the low number 1, and to make the logarithms, or the artificial numbers, as Napier had always called them, to increase with the natural numbers, instead of decreasing, which 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, which happened in 1618, 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, which was presently to appear. But although Napier had intimated in a note he had given in Wright's translation of the Canon Mirificum, as well as his Rabdologia, printed in 1617, that he intended to alter the scale, yet he altogether omits to state that Briggs either was the first to think of this improvement, or at least to publish it to the world. And as the same silence on this point was observed in Napier's posthumous work published in 1619 by his son, Briggs took occasion in the preface to his Arithmetica Logarithmica to assert his claims to the improvement he had now carried into execution.

The studied silence which Napier seems to have observed respecting the improvement of the system, which Briggs had communicated to him, has given just reason to suppose that he wished to be considered as the author of that improvement, as well as the original inventor. But although it is possible that he thought of it as soon as Briggs, it would seem to have been no more than justice, if, when announcing his intended change of the scale, he had acknowledged that the same idea had occurred to Briggs as well as to himself.

In 1620 Mr Edmund Gunter published his Canon of Triangles, which contains the artificial or logarithmic fines 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 fines and tangents that were published of this sort. Gunter also in 1623 reprinted the same in his book de Sectione et Radio, together with the Chilias prima of Briggs; and in the same year he applied the logarithms of numbers, fines, 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 first 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 the French language upon logarithms, and these 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 here gives the logarithms of 30000 natural numbers to fourteen places of figures, besides the index; namely, from 1 to 20000 and from 90000 to 100000, 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 70000 numbers, while he should employ his time upon the Canon of Logarithmic sines and tangents, and fo carry on both works at once.

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

Briggs himself lived also to complete a table of logarithmic sines and tangents, to the tenth 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 his death, which then happened, prevented him from completing the application and uses of them. However, when dying, he committed the performing of this office 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 the year 1633 under the care of Adrian Vlacq.

In the same year, 1633, Adrian Vlacq printed a work of his own, called Trigonometria Artificialis, sive Magnum Canon Triangulorum Logarithmorum ad Decadas Secundorum Scrupulorum Constructus. This work 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 20000 logarithms to ten places, besides the index, with their differences; and to the whole is prefixed a description of the tables and their applications, chiefly extracted from Briggs's Trigonometria Britannica, which we have already mentioned.

Gellibrand published also, 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 places of figures besides the index.

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

As it is of importance that such as have occasion to employ logarithms should know what works are held in estimation on account of their extent and accuracy, we shall enumerate the following.

1. 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 upon it when accuracy is required.

2. 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.

3. An edition of the same work, with some additions, printed in 1770 in Avignon in France. The tables in both editions are to seven places of figures.

4. Tables Portatives de Logarithmes, publiées à Londres, par Gardiner, augmentées et perfectionnées dans leur disposition, par M. Callet.—This work is most beautifully printed in a small octavo volume, and contains all the tables in Gardiner's quarto volume; with some additions and improvements.

5. Dr Hutton's Mathematical Tables, containing common hyperbolic and logistic logarithms, &c.—This work has passed through several editions, under the care of the learned author: it is perhaps the most common of any in this country, and is deservedly held in the highest estimation, both on account of its accuracy, and the very valuable information it contains respecting the history of logarithms, and other branches of mathematics connected with them.

6. Taylor's Table of Logarithmic Sines and Tangents to every second of the quadrant; to which is prefixed a table of logarithms from 1 to 100,000, &c.—This is a most valuable work; but being a very large quarto volume, and also very expensive, it is less adapted to general use than the preceding, which is an octavo, and may be had at a moderate price.

7. 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 le nouvelle Nature of nouvelle division centésimale de dix-mille-ieme en dix mil- Logarithms, lieme, &c. par Callet.—This work, which is in octavo, &c. may be reasonably expected to be very accurate, it being printed in the stereotype manner, by Didot.

In addition to these, it is proper that we should notice a stupendous work relating to logarithms, originally suggested by the celebrated Carnot, in conjunction with Prieur de la Côte d'Or, and Brunet de Montpelier, about the beginning of the French revolution. This enterprise was committed in the year 1794, to the care of Prony, a French 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 which had ever been executed or ever imagined*. It appears that two manuscript copies of the work were formed, composed of 17 volumes large folio; and containing, besides an introduction, the following tables.

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 24 decimals, in order to be published to 12 decimals and three columns of differences.

The printing of this work was begun at the expense of the French government, but was suspended at the fall of the affigats; whether it has been since resumed we cannot positively say, but it certainly is not yet completed.

SECT. I. OF THE NATURE OF LOGARITHMS AND THEIR CONSTRUCTION.

We have already shewn that the properties of logarithms are deducible from those of two series, the terms of one of which form a geometrical progression, and those of the other an arithmetical progression; and as this manner of treating the subject is simple, it is perhaps the best adapted of any to such of our readers as have not pursued the study of mathematics to any great extent. We shall now shew how, from the same principles, the logarithm of any proposed number whatever may be found.

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 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 Logarithms, determinate, and the logarithm of every number fixed &c. to one particular value.

Mr Briggs however 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 system, and hence the logarithms of the terms of the geometrical progression

1, 10, 100, 1000, 10000, &c.

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

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

That is, the logarithm of 1 being 0, and that of 10 being 1, the logarithm of 100 is 2, and 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 as 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; it is evident that 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 the number 5, according to Briggs's system.

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

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

Third step.—We have now obtained two numbers, namely Nature of namely 3.162277 and 5.623413, one on each side of Logarithms, 5, together with their logarithms 0.5 and .075, we therefore proceed exactly as before, and accordingly we find the geometrical mean, or \( \sqrt{(3.162277 \times 5.623413)} \), to be 4.216964, and the arithmetical mean, or \( \frac{0.5 + 0.75}{2} \) to be 0.625; therefore the logarithm of 4.216964 is 0.625.

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 its 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:

<table> <tr> <th>Numbers.</th> <th>Logarithms.</th> </tr> <tr><td>A = 1</td><td>0.000000</td></tr> <tr><td>B = 10</td><td>1.000000</td></tr> <tr><td>C = \( \sqrt{AB} = 3.162277 \)</td><td>0.500000</td></tr> <tr><td>D = \( \sqrt{BC} = 5.623413 \)</td><td>0.750000</td></tr> <tr><td>E = \( \sqrt{CD} = 4.216964 \)</td><td>0.625000</td></tr> <tr><td>F = \( \sqrt{DE} = 4.869674 \)</td><td>0.687500</td></tr> <tr><td>G = \( \sqrt{DF} = 5.232991 \)</td><td>0.718750</td></tr> <tr><td>H = \( \sqrt{FG} = 5.048065 \)</td><td>0.703125</td></tr> <tr><td>I = \( \sqrt{FH} = 4.958009 \)</td><td>0.6953125</td></tr> <tr><td>K = \( \sqrt{HI} = 5.002865 \)</td><td>0.6992187</td></tr> <tr><td>L = \( \sqrt{IK} = 4.980416 \)</td><td>0.6972656</td></tr> <tr><td>M = \( \sqrt{KL} = 4.991627 \)</td><td>0.6982421</td></tr> <tr><td>N = \( \sqrt{LM} = 4.997242 \)</td><td>0.6987304</td></tr> <tr><td>O = \( \sqrt{MN} = 5.000052 \)</td><td>0.6989745</td></tr> <tr><td>P = \( \sqrt{NO} = 4.998647 \)</td><td>0.6988525</td></tr> <tr><td>Q = \( \sqrt{OP} = 4.999350 \)</td><td>0.6989135</td></tr> <tr><td>R = \( \sqrt{OQ} = 4.999701 \)</td><td>0.6989440</td></tr> <tr><td>S = \( \sqrt{OR} = 4.999876 \)</td><td>0.6989192</td></tr> <tr><td>T = \( \sqrt{OS} = 4.999963 \)</td><td>0.6989668</td></tr> <tr><td>V = \( \sqrt{OT} = 5.000008 \)</td><td>0.6989707</td></tr> <tr><td>W = \( \sqrt{TV} = 4.999984 \)</td><td>0.6989687</td></tr> <tr><td>X = \( \sqrt{WV} = 4.999997 \)</td><td>0.6989697</td></tr> <tr><td>Y = \( \sqrt{VX} = 5.000003 \)</td><td>0.6989702</td></tr> <tr><td>Z = \( \sqrt{XY} = 5.000000 \)</td><td>0.6989700</td></tr> </table>

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

In performing the operations indicated in the preceding table, it will be necessary to find the geometrical means at the beginning to many more figures than are here put down, in order to obtain 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 Vlacq in the original construction of logarithms; but since the period in which they lived, others more early have been found, as we shall presently have occasion to explain.

The logarithm of any number whatever may be found by a series of calculations similar to that which we have just now 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 the logarithm of 5 is 0.6989700, therefore the logarithm of 2 is 0.3010300.

Having thus obtained the logarithms of 2 and 5, in addition to those of 10, 100, 1000, &c. we may thence find the logarithms of innumerable other numbers. Thus, because 4 = 2 × 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 × 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 × 4, of 16 = 2 × 8, of 25 = 5 × 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 definition of a compound ratio, as laid down by writers on geometry, if there be any number of magnitudes A, B, C, D, which are continual proportionals, or such that the ratio of A to B is equal to the ratio of B to C, and that again is equal to the ratio of C to D, and so on, the ratio of the first of these magnitudes A to the third C is considered as made up of two equal ratios, each equal to 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 up of three equal ratios, each equal to the ratio of the first to the second, and so on. (See GEOMETRY, Sect. III. Def. 10, 11, and 12.) 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, (GEOMETRY, Sect. III. Def. 4,) the ratio of 81 to 3 will be triple the ratio of 9 to 3; and in like manner the ratio of 27 to 3 will be double the ratio 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 the ratio 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 respect of their magnitudes, in the same manner as we compare lines or quantities of any kind whatever. Nature of whatever. And as when estimating the relative magnitudes of two quantities, two lines for example, if we find that 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.

It is well known that there may be lines and other quantities, which, as they admit of no common measure, are said to be incommensurable to each other; and the same will also 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 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 any number of 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 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 1000000: then, as each of these will be very near to the ratio of equality, (for it will be the ratio of the first term to the second of a series 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 1000000; then, as it may be proved that the ratio of 2 to 1 will be very nearly the same as a ratio composed of 301030 of these; and that the ratio of 3 to 1 will be nearly equal to a ratio composed of 477121 of them, and that the ratio of 4 to 1 will be nearly equal to a ratio composed of 602060 of them, and so on; these numbers, viz. 1000000, 301030, 477121, and 602060, 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 λογισμός ἀριθμοῦ, 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 1000000 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 301030 and 477121 respectively; as, however, these measures are not absolute, but relative quantities, we may assume any other numbers whatever instead of these, provided they have the same proportions to each other as these numbers have among themselves. 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 301030, 477121, &c. will be .301030 and .477121, &c. respectively, that is, each will be the 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 of the subject, chose the number 2.718282; and hence it happens that the logarithms of the ratios are expressed by different numbers in the two systems.

It yet remains for us to shew the identity of the properties of logarithms, as explained in the two different views we have now given of the subject; and this may be done as follows.

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 \(A \times B\) to B, and of B to 1; (see Geometry, Part III. Def. 10.) but since \(A \times B\), B, A, and 1 are four proportionals, the ratio of \(A \times B\) to B is equal to the ratio of A to 1. Therefore 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 be equal to 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.

And because log. \((A \times B)\)=log. A+log. B, therefore, log. B=log. \((A \times B)\)—log. A. In this equation 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. \(\frac{C}{D}=\log. C-\log. D\).

We have now given 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 according to the strictest rules of geometrical reasoning; and in this he has been followed by Mercator, Halley, Cotes, as well as by other mathematicians of later times, as by Mr Baron Maferes, in his "Elements of Plane Trigonometry," a work in which the whole theory of logarithms is treated with all that perspicuity and accuracy which characterize the ingenious author's various writings. The same mode of treating the subject was likewise adopted by that excellent geometrician Dr Robert Simson, as appears by a short tract in Latin, written by him and published in his posthumous works. As, however, the doctrine of ratios is of a very abstract nature, 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 Nature of Logarithms, &c.

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.

If we attend to the common scale of notation in arithmetic, we shall find that it is so contrived as to express all numbers whatever by means of the powers of the number 10, which is the root of the scale, and the nine digits which serve as coefficients 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 471509 is otherwise expressed by \( 4R^2 + 7R^4 + 1R^3 + 5R^2 + 0R^1 + 9R^0 \), which is equivalent to \( 4R^2 + 7R^4 + 1R^3 + 5R^2 + 9 \). Again, the mixt number 371.243 is expressed by \( 3R^2 + 7R^4 + R^0 + 2R^{-1} + \frac{4}{R^3} + \frac{3}{R^2} \), or by \( 3R^2 + 7R^4 + R^0 + 2R^{-1} + 4R^{-2} + 3R^{-3} \).

As to vulgar fractions, by transforming them to decimals, they may be expressed in the same manner. Thus \( \frac{1}{3} = .375 = 3R^{-1} + 7R^{-2} + 5R^{-3} \). Also \( \frac{3}{7} = .666, \&c. = 6R^{-1} + 6R^{-2} + 6R^{-3} + \&c. \)

Although the number 10 has been fixed upon as the root of the scale of notation, any other number may be employed to express all numbers whatever in the same manner; and some numbers are even preferable to 10. Thus, making 8 the root of a scale, and denoting it by R, the number 2735, when expressed according to this scale, is \( 5R^1 + 2R^2 + 5R^1 + 7R^2 \), or \( 5R^1 + 2R^2 + 5R^1 + 7 \); and here we may observe, that if a number greater than 10 were assumed as the root of the scale of notation, it would be necessary to adopt some new numeral characters in addition to those in common use, and if a smaller number were assumed, we might dispense with some of those we already have.

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

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

\[ \begin{align*} 1 &= 2^0 \\ 2 &= 2^1 \\ 3 &= 2^{1.58496}, \text{ nearly} \\ 4 &= 2^2 \\ 5 &= 2^{2.3329}, \text{ nearly} \\ 6 &= 2^{2.8806}, \text{ nearly} \\ 7 &= 2^{2.8673}, \text{ nearly} \\ 8 &= 2^3 \\ 9 &= 2^{3.1699} \\ 10 &= 2^{3.3219}, \text{ nearly}, \end{align*} \]

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

\[ \begin{align*} 1 &= 10^0 \\ 2 &= 10^{0.30103} \\ 3 &= 10^{0.47712} \\ 4 &= 10^{0.60206} \\ 5 &= 10^{0.69907} \\ 6 &= 10^{0.77855} \\ 7 &= 10^{0.84510} \\ 8 &= 10^{0.90309} \\ 9 &= 10^{0.95424} \\ 10 &= 10^1. \end{align*} \]

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

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

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

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

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

From these properties it is easy to see in what manner a table that exhibits 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 equal to 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 into one sum, taking them from the table, and to look in the table for the number whose logarithm is equal to that sum, and this number will be the product required. Also, because the excess of the logarithm of the dividend above that of the divisor is equal to the logarithm of the quotient; as often as we have occasion to divide one number by another, we have only.

Nature of 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 shall find the quotient.

As the logarithm of any power of a number is the product of the logarithm of the number, and 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, in this case \( r^A = 1 \); but this equation can only be satisfied by putting \( A = 0 \). 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 \( r \) to be a positive number greater than unity, and \( a \) a positive number greater than unity, then A will be a positive number; for if it be negative we would have \( a = r^{-\lambda} = \frac{1}{r^\lambda} \) a proper fraction, and at the same time a number greater than unity by hypothesis, which is impossible. If on the contrary we suppose \( a \) a proper fraction, then A must necessarily be negative, for if it were positive, then \( r^A \) 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 impossible. Therefore, in every system, the base of which is greater than unity, the logarithm of a whole or mixt number is always positive, but the logarithm of a proper fraction is always negative.

Because the logarithm of \( r \) is unity, the logarithm of \( r^n \) will be \( n \); therefore, the logarithm of any integer 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 taken according to the last. Then because \( a = r^A \), and \( a = r'^{A'} \), it follows that \( r^A = r'^{A'} \), and \( r = r'^{\frac{A'}{A}} \). Let us now suppose that \( r'' \) is the base of a third system of logarithms, and R and R' the logarithms of \( r \) and \( r' \) taken according to this third system; then because

\[ r''^R = r, \quad r''^{R'} = r'; \] we have \( r''^{RR'} = r', \quad r''^{R'R} = r''^R \); therefore \( r''^R = r''^{R'} \), and \( r'' = r''^{\frac{R'}{R}} \); but we have already found \( r'' = r^A \), therefore \( r''^A = r''^{\frac{R'}{R}} \), and consequently

\[ \frac{A'}{A} = \frac{R'}{R} \] and \( A : A' :: R : R' \) :: \( \frac{1}{R} : \frac{1}{R'} \).

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, taken according to any system whatever.

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

Having now explained the properties which belong to the logarithms of any system whatever, 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 we have this equation

\[ y = r^x. \]

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

\[ 1 + x a + \frac{x(x-1)}{1 \cdot 2} a^2 + \frac{x(x-1)(x-2)}{1 \cdot 2 \cdot 3} a^3, \] \[ + \frac{x(x-1)(x-2)(x-3)}{1 \cdot 2 \cdot 3 \cdot 4} a^4 + \text{&c.} \]

Let this series, the terms of which are arranged according to the powers of the quantity \( a \), be transformed into another, the terms of which 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,

\[ 1 = 1, \] \[ x a = -a x, \] \[ \frac{x(x-1)}{1 \cdot 2} a^2 = -\frac{a^3}{2} x + \frac{a^2}{2} x^2, \] \[ \frac{x(x-1)(x-2)}{1 \cdot 2 \cdot 3} a^3 = -\frac{a^4}{3} x + \frac{a^3}{2} x^2 - \frac{a^2}{6} x^3, \] \[ \frac{x(x-1)(x-2)(x-3)}{1 \cdot 2 \cdot 3 \cdot 4} a^4 = -\frac{a^5}{4} x + \frac{11 a^4}{24} x^2 - \frac{a^3}{4} x^3 + \frac{a^2}{24} x^4, \] \& c.

fo that adding into one sum the quantities on each side of the sign \( = \), and recollecting that the sum of these on the left-hand side is equal to \( y \), we have Nature of Logarithms, &c.

\[ y = r^x = \left\{ \begin{array}{l} 1 \\ + (a - \frac{a^2}{2} + \frac{a^3}{3} - \frac{a^4}{4} + \text{&c.})x, \\ + (\frac{a^2}{2} - \frac{a^3}{2} + \frac{11a^4}{24} - \text{&c.})x^2, \\ + (\frac{a^3}{6} - \frac{a^4}{4} + \text{&c.})x^3, \\ + (\frac{a^4}{24} + \text{&c.})x^4, \\ + \text{&c.} \end{array} \right. \]

which equation, by substituting

A for \( a - \frac{a^2}{2} + \frac{a^3}{3} - \frac{a^4}{4} + \text{&c.} \)

\( A' \) for \( \frac{a^2}{2} - \frac{a^3}{2} + \frac{11a^4}{24} - \text{&c.} \)

\( A'' \) for \( \frac{a^3}{6} - \frac{a^4}{4} + \text{&c.} \)

\( A''' \) for \( \frac{a^4}{24} + \text{&c.} \)

may be abbreviated to

\[ r^x = 1 + Ax + A'x^2 + A''x^3 + A'''x^4 + \text{&c.} \]

Next, to determine the law of connexion of the quantities A, \( A' \), \( A'' \), \( A''' \), &c. let \( x + zx \) be substituted in the last equation for \( x \), (here \( z \) is put for any indefinite quantity) thus it becomes

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

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

\[ r^x = 1 + Ax + A'x^2 + A''x^3 + A'''x^4 + \text{&c.} \]

for the very same reason

\[ r^{x+z} = 1 + Az + A'z^2 + A''z^3 + A'''z^4 + \text{&c.} \]

therefore the series

\[ 1 + A(x+z) + A'(x+z)^2 + A''(x+z)^3 + A'''(x+z)^4 + \text{&c.} \]

is equal to the product of the two series

\[ \begin{align*} 1 + Ax + A'x^2 + A''x^3 + A'''x^4 + \text{&c.} \\ 1 + Az + A'z^2 + A''z^3 + A'''z^4 + \text{&c.} \end{align*} \]

That is, by actual involution and multiplication

\[ \begin{align*} 1 + Ax + A'x^2 + A''x^3 + A'''x^4 + \text{&c.} \\ + Az + 2A'xz + 3A''x^2z + 4A'''x^3z + \text{&c.} \\ + A'z^2 + 3A''xz^2 + 6A'''x^2z^2 + \text{&c.} \\ + A''z^3 + 4A'''xz^3 + \text{&c.} \\ + A'''z^4 + \text{&c.} \end{align*} \]

\[ = \left\{ \begin{array}{l} 1 + Ax + A'x^2 + A''x^3 + A'''x^4 + \text{&c.} \\ + Az + A'xz + AA'x^2z + AA''x^3z + \text{&c.} \\ + A'z^2 + AA''x^2z^2 + AA'''x^3z^2 + \text{&c.} \\ + A''z^3 + AA'''x^2z^3 + \text{&c.} \\ + A'''z^4 + \text{&c.} \end{array} \right. \]

Now as the quantities A, \( A' \), \( A'' \), &c. are quite independent of \( x \) and \( z \), the two sides of the equation can only be identical upon the supposition that the coefficients of like terms in each are equal; therefore, setting aside the first line of each side of the equation, because their terms are the same, and also the first term of the second line, for the same reason, let the coefficients of the remaining terms be put equal to one another, thus we have

\[ \begin{cases} A' = 2A \\ AA' = 3A' \\ AA'' = 4A''' \\ \text{&c.} \end{cases} \]

and hence

\[ \begin{cases} A' = \frac{A^2}{1 \cdot 2} \\ A'' = \frac{A^3}{1 \cdot 2 \cdot 3} \\ A''' = \frac{A^4}{1 \cdot 2 \cdot 3 \cdot 4} \\ \text{&c.} \end{cases} \]

Here the law of the coefficients A, \( A' \), \( A'' \), &c. is obvious, each being formed from the preceding by multiplying it by A, and dividing by the exponent of the power of A which is thus formed. Let these values of \( A' \), \( A'' \), &c. be now substituted in the equation

\[ y = r^x = 1 + Ax + A'x^2 + A''x^3 + \text{&c.} \]

and it becomes,

\[ y = 1 + Ax + \frac{A^2}{1 \cdot 2}x^2 + \frac{A^3}{1 \cdot 2 \cdot 3}x^3 + \frac{A^4}{1 \cdot 2 \cdot 3 \cdot 4}x^4 + \text{&c.} \]

thus we have obtained a general formula expressing a number in terms of its logarithm and the base of the system, for we must recollect that the quantity A which is equal to

\[ a - \frac{a^2}{2} + \frac{a^3}{3} - \frac{a^4}{4} + \frac{a^5}{5} - \text{&c.} \]

is otherwise expressed by

\[ r - 1 - \frac{(r-1)^2}{2} + \frac{(r-1)^3}{3} - \frac{(r-1)^4}{4} + \frac{(r-1)^5}{5} - \text{&c.} \]

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

If in the formula

\[ r^x = 1 + Ax + \frac{A^2}{1 \cdot 2}x^2 + \frac{A^3}{1 \cdot 2 \cdot 3}x^3 + \frac{A^4}{1 \cdot 2 \cdot 3 \cdot 4}x^4 + \text{&c.} \]

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

\[ r = 1 + A + \frac{A^2}{1 \cdot 2} + \frac{A^3}{1 \cdot 2 \cdot 3} + \frac{A^4}{1 \cdot 2 \cdot 3 \cdot 4} + \text{&c.} \]

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

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

\[ \frac{1}{r^x} = 1 + \frac{1}{1 \cdot 2} + \frac{1}{1 \cdot 2 \cdot 3} + \frac{1}{1 \cdot 2 \cdot 3 \cdot 4} + \text{&c.} \]

Thus

(Vol. XII. Part I.)

(a) For other analytic methods of investigating the same formula, see Algebra, § 293, and Fluxions, § 54. and § 70. Ex. 1. also § 200. Prob. 1. Thus the quantity \( r^{\frac{1}{A}} \), whatever be the value of \( r \), is evidently equal to a constant number, which, as appears from the last equation, is equal to the value of \( r \) when \( A = 1 \). By adding together a sufficient number of the terms of the series expressing the value of \( r^{\frac{1}{A}} \), we find that quantity equal to

\[ 2.718281828459045 \ldots \]

Let this number be denoted by \( e \), and we have \( r^{\frac{1}{A}} = e \), and \( r = e^A \); hence it appears, that if the number \( e \) be considered as the base of a logarithmic system, the quantity \( A \), that is

\[ r - 1 - \frac{(r-1)^2}{2} + \frac{(r-1)^3}{3} - \frac{(r-1)^4}{4} + \frac{(r-1)^5}{5} - \text{&c.} \]

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

\[ y - 1 - \frac{(y-1)^2}{2} + \frac{(y-1)^3}{3} - \frac{(y-1)^4}{4} + \frac{(y-1)^5}{5} - \text{&c.} \]

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 shewn that the logarithms of the same number, taken according to two different systems, are to each other as the reciprocals of the logarithms of the bases of the systems, these last logarithms being taken according to any system whatever, that is,

\[ \log.\ y \text{ to base } e : \log.\ y \text{ to base } r :: \frac{1}{\log.\ e} : \frac{1}{\log.\ r}; \]

hence we find

\[ \log.\ y \text{ to base } r = \frac{\log.\ e}{\log.\ r} \times \log.\ y \text{ to base } e. \]

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

\[ \log.\ y = \frac{\log.\ e}{\log.\ r} \left\{ y - 1 - \frac{(y-1)^2}{2} + \frac{(y-1)^3}{3} - \frac{(y-1)^4}{4} + \text{&c.} \right\} \]

which is a general formula for the logarithm of any number whatever, to the base \( r \). And it is to be remembered 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 (b).

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

\[ \log.\ y = y - 1 - \frac{(y-1)^2}{2} + \frac{(y-1)^3}{3} - \frac{(y-1)^4}{4} + \text{&c.} \]

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

As the constant multiplier \( \frac{\log.\ e}{\log.\ r} \), 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 Napierian logarithm of \( r \). If however we suppose the logarithms taken to the base \( r \), then the numerator will be \( \log.\ e \) to base \( r \); and the denominator will be unity, so that the modulus of any system whose base is \( r \), is the reciprocal of the Napierian logarithm of that base; or it is the logarithm of the number \( e \) (the base of the Napierian system) to the base \( r \).

In the Napierian system the modulus is unity, and hence the logarithms of this system, as far as depends upon facility of computation, are the most simple of any. It was, however, soon found that a system whose base should be the same as the root of the scale of the arithmetical notation, viz. the number 10, would be the most convenient of any in practice; and accordingly such a system was actually constructed by Mr Briggs. This is the only one now in common use, and is called Briggs's system, also the common system of logarithms. The modulus of this system therefore is the reciprocal of the Napierian logarithm of 10; or it 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

\[ \log.\ y = M \left\{ 1 - y - \frac{(1-y)^2}{2} + \frac{(1-y)^3}{3} - \frac{(1-y)^4}{4} + \text{&c.} \right\} \]

If the number \( y \), whose logarithm is required be very near 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 it \( \sqrt[n]{y} \) instead of \( y \), and observing that \( \log.\ (\sqrt[n]{y}) = \frac{\log.\ y}{n} \);

Nature of Logarithms, &c.

it thus becomes

\[\log y = nM \left\{ n\sqrt{y-1} - \frac{1}{2}(n\sqrt{y-1})^3 + \frac{1}{3}(n\sqrt{y-1})^5 - \ldots \right\}\]

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

If we suppose \(n\) to be taken negative, then \(n\sqrt{y} = \frac{I}{-\sqrt{y}}\), and the series which expresses log. \(y\) becomes, by changing the signs,

\[\log y = nM \left\{ I - \frac{I}{n\sqrt{y}} + \frac{1}{2}\left(1 - \frac{I}{n\sqrt{y}}\right)^2 + \frac{1}{3}\left(1 - \frac{I}{n\sqrt{y}}\right)^3 + \ldots \right\},\]

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 \(y\) being greater than unity, \(n\sqrt{y}\) will also be greater than unity, and \(y\) being less than unity, \(n\sqrt{y}\) will also be less than unity, 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 \(M = \frac{1}{\text{Nap. log. .10}}\), therefore Nap. log. .10

\[\frac{I}{M} = \frac{1}{\text{Nap. log. .10}};\] hence by the two last formulas we have

\[\frac{I}{M} = n \left\{ n\sqrt{.10-1} - \frac{1}{2}(n\sqrt{.10-1})^3 + \frac{1}{3}(n\sqrt{.10-1})^5 - \ldots \right\}\]

also

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

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 as \(n\sqrt{y}-1\) is a fraction less than unity, the expression

\[\frac{n}{n\sqrt{y}}\]

will also be less than unity, seeing that it is equal to \(\frac{\sqrt{y}-1}{n\sqrt{y}}\). 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, that is a quantity which is to be subtracted from the first, that we may have the value of log. \(y\). Hence we may infer that

\[\log y < n M (n\sqrt{y}-1).\]

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, that is, a quantity which must be added to the first to give the value of log. \(y\); so that we have

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

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.

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

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

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

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 \(m\) cyphers between the decimal point and the first significant figure on the right. So that in general, as the error is the smaller according as \(n\) the exponent of the root is greater, we may conclude that it becomes nothing, or may be reckoned as nothing, when \(n\) is taken indefinitely great; and this being the case, we may conclude that either of these expressions

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

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 cyphers 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 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 equal to

\[\frac{I}{I} \times \frac{I}{n\sqrt{.10-I}},\]

nearly,

when \(n\) is some very great number. Let us suppose \(n=2^{80}=8^n\); then, dividing unity by 8, and this result again by 8, and so on, we shall after 20 divisions have 1/7 or \(1/8^{20}\) equal to

\(0.000000000000086\ 73617\ 37988\ 40354\).

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 \(n\sqrt{.10}\) equal to

\(1.0000000000019971742081255052703251\).

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

\[ \frac{8673617398849354}{199717420812550527} = 0.4342944819. \]

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

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

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

\[ 1.00000000000000095289426407458932. \]

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

\[ \log. 3 = \frac{\sqrt{3}-1}{\sqrt{10}-1} = \frac{95289426407458932}{199717420812550527} = .477121154719662. \]

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 \( \sqrt{y} \) a value which is unity followed by a decimal fraction; and then \( \sqrt{y-1} \), being a fraction, its powers will also be fractions, which will be so much the smaller as their exponents are greater; thus a certain number of terms of the series will serve to express the logarithm to as many decimal places as may be required.

There are yet other analytical artifices by which the series

\[ \log.y = M \left\{ y - 1 - \frac{1}{2}(y-1)^2 + \frac{1}{3}(y-1)^3 - \frac{1}{4}(y-1)^4 + &c. \right\} \]

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

\[ \log.(1+u) = M \left( u - \frac{u^2}{2} + \frac{u^3}{3} - \frac{u^4}{4} + \frac{u^5}{5}, &c. \right). \]

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

\[ \log.(1-u) = M \left( -u - \frac{u^2}{2} + \frac{u^3}{3} - \frac{u^4}{4} + \frac{u^5}{5}, &c. \right). \]

Let each side of the latter equation be subtracted from the corresponding side of the former; the result on the left-hand side will be \( \log.(1+u) - \log.(1-u) \), which, by the nature of logarithms, is equal to \( \log.\frac{1+u}{1-u} \); and on the right-hand side the alternate terms of the two series, having the same sign, these will by subtraction destroy each other, so that we shall have

\[ \log.\frac{1+u}{1-u} = 2M \left\{ u + \frac{u^3}{3} + \frac{u^5}{5} + \frac{u^7}{7}, &c. \right\} \]

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

\[ \log.x = 2M \left\{ \frac{x-1}{x+1} + \frac{1}{2} \left( \frac{x-1}{x+1} \right)^3 + \frac{1}{3} \left( \frac{x-1}{x+1} \right)^5 + &c. \right\}; \]

and this formula for the logarithm of a number is not only simple, but has also the property of converging in every possible case.

That we may give an example of the utility of this formula, we shall employ it in the calculation of the Napierian logarithm of 2, which by the above formula will be

\[ 2 \left( \frac{1}{3} + \frac{1}{3 \cdot 3^3} + \frac{1}{5 \cdot 3^5} + \frac{1}{7 \cdot 3^7} + \frac{1}{9 \cdot 3^9} + &c. \right) = A + \frac{1}{3}B + \frac{1}{3^3}C + \frac{1}{3^5}D + \frac{1}{3^7}E + &c. \]

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

\[ \begin{align*} A &= .666666666666 \\ \frac{1}{3}B &= .074074974074 \\ \frac{1}{3^3}C &= .008230452674 \\ \frac{1}{3^5}D &= .000914494742 \\ \frac{1}{3^7}E &= .0000101610527 \\ \frac{1}{3^9}F &= .0000011290059 \\ \frac{1}{3^{11}}G &= .0000001254451 \\ \frac{1}{3^{13}}H &= .0000000139383 \\ \frac{1}{3^{15}}I &= .0000000015487 \\ \frac{1}{3^{17}}K &= .0000000001721 \\ \frac{1}{3^{19}}L &= .0000000000191 \\ \frac{1}{3^{21}}M &= .0000000000021 \end{align*} \]

\[ \begin{align*} A &= .666666666666 \\ \frac{1}{3}B &= .024601358025 \\ \frac{1}{3^3}C &= .001646000535 \\ \frac{1}{3^5}D &= .000130642106 \\ \frac{1}{3^7}E &= .000011290059 \\ \frac{1}{3^9}F &= .000001026369 \\ \frac{1}{3^{11}}G &= .000000096496 \\ \frac{1}{3^{13}}H &= .000000009292 \\ \frac{1}{3^{15}}I &= .000000000911 \\ \frac{1}{3^{17}}K &= .000000000091 \\ \frac{1}{3^{19}}L &= .0000000000091 \\ \frac{1}{3^{21}}M &= .0000000000001 \end{align*} \]

Nap. log. 2 = .693147180551

Thus, by a very easy calculation, we have obtained the Napierian logarithm of 2 true to the first ten places of figures; the accurate value, as far as the 12th place, being 0.693147180550.

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 have found the numerical value of M, and the common logarithm of 3, must be very apparent.

In the same manner as we have found the logarithm of 2 we may find those of 3, 5, &c. In computing the logarithm Nature of 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 \( \frac{5-1}{5+1} = \frac{1}{3} \); 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 \( \frac{1+u}{1-u} = 1 + \frac{\infty}{n^2} \), then, by resolving this equation in respect of \( u \), we have \( u = \frac{\infty}{2n+\infty} \). Let these values of \( \frac{1+u}{1-u} \) and \( u \) be substituted in the formula,

\[ \log. \frac{1+u}{1-u} = 2M \left( u + \frac{u^3}{3} + \frac{u^5}{5} + \frac{u^7}{7} + &c. \right) \]

and we have \( \log. \left( 1 + \frac{\infty}{n} \right) \) equal to

\[ 2M \left\{ \frac{\infty}{2n+\infty} + \frac{1}{3} \left( \frac{\infty}{2n+\infty} \right)^3 + \frac{1}{5} \left( \frac{\infty}{2n+\infty} \right)^5 + &c. \right\} \]

but \( \log. \left( 1 + \frac{\infty}{n} \right) = \log. \frac{n+\infty}{n} = \log. (n+\infty) - \log. n \),

therefore, by substituting this value of \( \log. \frac{n+\infty}{n} \), and transposing \( \log. n \) to the other side of the equation, we have

\[ \log. (n+\infty) = \log. n + 2M \left\{ \frac{\infty}{2n+\infty} + \frac{1}{3} \left( \frac{\infty}{2n+\infty} \right)^3 + \frac{1}{5} \left( \frac{\infty}{2n+\infty} \right)^5 + &c. \right\}. \]

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, which has been already found. We have here \( n = 2 \), \( \infty = 1 \), and \( \frac{\infty}{2n+\infty} = \frac{1}{3} \). Therefore the logarithm required is equal to

\[ \log. 2 + 2 \left( \frac{1}{5} + \frac{1}{3.5^3} + \frac{1}{5.5^5} + \frac{1}{7.5^7} + &c. \right) = \log. 2 + A + \frac{1}{5} B + \frac{1}{5} C + \frac{1}{5} D + \frac{1}{5} E + &c. \]

where \( A \) is put for \( \frac{1}{5} \), \( B \) for \( \frac{A}{2.5} \), \( C \) for \( \frac{B}{2.5} \), and so on.

The calculation may stand thus:

\[ \begin{align*} A &= .4000000000 \\ B &= \frac{1}{5} A = .0800000000 \\ C &= \frac{1}{5} B = .0160000000 \\ D &= \frac{1}{5} C = .0032000000 \\ E &= \frac{1}{5} D = .0006400000 \\ F &= \frac{1}{5} E = .0001280000 \\ G &= \frac{1}{5} F = .0000256000 \\ H &= \frac{1}{5} G = .0000051200 \\ &\vdots \end{align*} \]

Nature of logarithms, &c.

\[ \begin{align*} E &= \frac{1}{2.5} D = .00001024000 \\ F &= \frac{1}{2.5} E = .00000204960 \\ G &= \frac{1}{2.5} F = .00000040960 \\ H &= \frac{1}{2.5} G = .000000081920 \\ &\vdots \end{align*} \]

\[ \begin{align*} A &= .4000000000 \\ B &= .0803333333 \\ C &= .0012800000 \\ D &= .0000357143 \\ E &= .00000113778 \\ F &= .00000003724 \\ G &= .00000000125 \\ H &= .00000000004 \\ &\vdots \end{align*} \]

.405465108108 Nap. log. 2 = .693147180351 Nap. log. 3 = 1.098612288659

This logarithm is true to 10 decimal places, the accurate value to 12 figures being 1.098612288668.

To find the Napierian logarithm of 4. This is immediately had from that of 2 by considering that as \( 4 = 2^2 \), therefore \( \log. 4 = \log. 2 + \log. 2 \).

Nap. log. 2 = .693147180351 Nap. log. 4 = 1.386294361102

This logarithm is also true to 10 places besides the integer.

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

\[ \log. 4 + 2 \left( \frac{1}{9} + \frac{1}{3.9^3} + \frac{1}{5.9^5} + \frac{1}{7.9^7} + &c. \right) \] \[ \log. 4 + A + \frac{1}{5} B + \frac{1}{5} C + \frac{1}{5} D + &c. \] where \( A = \frac{1}{9} \), \( B = \frac{1}{3.9^3} \), \( C = \frac{1}{5.9^5} \), \( D = \frac{1}{7.9^7} \), &c.

The calculation.

\[ \begin{align*} A &= .222222222222 \\ B &= \frac{1}{3.9} A = .002743484225 \\ C &= \frac{1}{5.9^5} B = .000033870176 \\ D &= \frac{1}{7.9^7} C = .000000418150 \\ E &= \frac{1}{9.9^9} D = .00000005162 \\ F &= \frac{1}{11.9^{11}} E = .00000000064 \\ &\vdots \end{align*} \]

A = .222222222222 \( \frac{1}{3} \) B = .00014494742 \( \frac{1}{5} \) C = .000006774035 \( \frac{1}{7} \) D = .000000059736 \( \frac{1}{9} \) E = .000000000574 \( \frac{1}{11} \) F = .000000000006

.223143551315 Nap. log. 4 = 1.386294361102

Nap. log. 5 = 1.609437912417

This result is also correct to the first ten places of decimals.

The logarithm of 6 is found from those of 2 and 3 by considering, that because \( 6 = 2 \times 3 \), therefore \( \log_6 6 = \log_2 2 + \log_3 3 \).

\[ \begin{align*} \text{Nap. log. } 2 &= 0.693147180551 \\ \text{Nap. log. } 3 &= 1.098612288659 \\ \text{Nap. log. } 6 &= 1.791759469210 \end{align*} \]

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

We might find the logarithm of 7 from the logarithm 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 by reasoning thus. Because

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

\[ \log_7 7 = \frac{1}{2} \log_2 2 + \log_5 5 - \frac{1}{2} \log_7 \frac{50}{49} \]

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

\[ \log x = 2M \left\{ \frac{x-1}{x+1} + \frac{1}{3} \left( \frac{x-1}{x+1} \right)^3 + \frac{1}{5} \left( \frac{x-1}{x+1} \right)^5 + &c. \right\} \]

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

\[ \begin{align*} \text{Nap. log. } \frac{50}{49} &= 2 \left( \frac{1}{99} + \frac{1}{3.99^3} + \frac{1}{5.99^5} + &c. \right) \\ &= A + \frac{1}{3} B + \frac{1}{5} C + &c. \end{align*} \]

where \( A = \frac{2}{9.11} \), \( B = \frac{A}{9^2.11^2} \), \( C = \frac{B}{9^3.11^3} \), &c. This series converges with great rapidity, and a few of its terms will be sufficient to give the logarithm of 7, as appears from the following operation.

\[ \begin{align*} A &= .020202020202 \\ B &= \frac{1}{9^2.11^2} A = .000002061220 \\ C &= \frac{1}{9^3.11^3} B = .000000000210 \end{align*} \]

\[ \begin{align*} A &= .020202020202 \\ \frac{1}{3} B &= .000000687073 \\ \frac{1}{5} C &= .000000000042 \end{align*} \]

\[ \text{Nap. log. } \frac{50}{49} = .020202707317 \]

\[ \begin{align*} \frac{1}{2} \log_2 2 &= 0.346573590275 \\ \log_5 5 &= 1.690437912417 \\ &= 1.956011502692 \\ \frac{1}{2} \log_7 \frac{50}{49} &= 0.010101353658 \\ \text{Nap. log. } 7 &= 1.945910149034 \end{align*} \]

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:

\[ \begin{align*} \text{Nap. log. } 2 &= 0.693147180551 \\ \text{Nap. log. } 3 &= 1.098612288659 \\ \text{Nap. log. } 8 &= 2.079441541653 \\ \text{Nap. log. } 9 &= 2.197224577318 \\ \text{Nap. log. } 2 &= 0.693147180551 \\ \text{Nap. log. } 5 &= 1.690437912417 \\ \text{Nap. log. } 10 &= 2.302585092968 \end{align*} \]

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

\[ \frac{1}{2.302585092968} = .4342944819 \]

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.

\[ \begin{align*} M &= .434294481903251827651128918917 \\ \frac{1}{M} &= 2.302585092994045684017991454684 \end{align*} \]

The formulas we have 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

\[ \log x = 2M \left\{ \frac{x-1}{x+1} + \frac{1}{3} \left( \frac{x-1}{x+1} \right)^3 + \frac{1}{5} \left( \frac{x-1}{x+1} \right)^5 + &c. \right\} \]

If we now suppose

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

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

\[ \log \frac{n^2}{(n-1)(n+1)} = 2M \left\{ \frac{1}{2n^2-1} + \frac{1}{3} \left( \frac{1}{2n^2-1} \right)^3 + \frac{1}{5} \left( \frac{1}{2n^2-1} \right)^5 + &c. \right\} \]

But

Nature of Logarithms. But \( \log \frac{n^2}{(n-1)(n+1)} = 2 \log n - \log (n-1) - \log (n+1) \), therefore, putting N for the series

\[ 2M \left\{ \frac{1}{2n^2-1} + \frac{1}{3} \left( \frac{1}{2n^2-1} \right)^3 + \frac{1}{5} \left( \frac{1}{2n^2-1} \right)^5 + &c. \right\} \]

we have this formula,

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

and hence, as often as we have the logarithms of any two of three numbers whose common difference is unity, the logarithm of the remaining number may be found. Example. Having given

the common log. of 9 = 0.95424250943 the common log. of 10 = 1;

it is required to find the common logarithm of 11.

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

\[ \log 11 = 2 \log 10 - \log 9 - N, \]

where \( N = \frac{2M}{199} + \frac{2M}{3.199^3} + &c. \)

\( M \) being .43429448190.

Calculation of N.

\[ A = \frac{2M}{199} = .00436476866 \\ B = \frac{A}{3.199^2} = .0000003674 \\ \] \[ \frac{.00436480540}{2 \log 10 = 2.00000000000} \] \[ \log 9 = 0.95424250943 \\ N = 0.00436480540 \] \[ \log 9 + N = 0.95860731483 \] \[ \log 11 = 1.04139268517 \]

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

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

that is, substituting \( \log 9 + \log 11 \) for \( \log 99 \), \( \log 2 + 2 \log 7 \) for \( \log 98 \), and \( 2 \log 10 \) for \( \log 100 \),

\[ 2 \log 9 + 2 \log 11 - \log 2 - 2 \log 7 - 2 \log 10 = N, \]

and hence by transposition, &c.

\[ \log 11 = \frac{1}{3} N + \frac{1}{3} \log 2 + \log 7 - \log 9 + \log 10; \]

and in this equation.

\[ N = \frac{2M}{19601} + \frac{2M}{19601^3} + &c. \]

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

\[ \log x = 2M \left\{ \frac{x-1}{x+1} + \frac{1}{3} \left( \frac{x-1}{x+1} \right)^3 + \frac{1}{5} \left( \frac{x-1}{x+1} \right)^5 + &c. \right\} \]

Let us assume

\[ x = \frac{(n-1)^2(n+2)}{(n-2)(n+1)^2} = \frac{n^3 - 3n + 2}{n^3 - 3n - 2} \]

then

\[ \frac{x-1}{x+1} = \frac{2}{n^3 - 3n}. \]

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

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

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

\[ 2M \left\{ \frac{2}{n^3 - 3n} + \frac{1}{3} \left( \frac{2}{n^3 - 3n} \right)^3 + \frac{1}{5} \left( \frac{2}{n^3 - 3n} \right)^5 + &c. \right\} \]

we have this formula,

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

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

\[ \log 7 + 2 \log 2 - 3 \log 3 = \frac{2M}{55} + \frac{2M}{3.55^3} + &c. \] \[ -2 \log 7 + \log 2 + 2 \log 5 = \frac{2M}{99} + \frac{2M}{3.99^3} + &c. \] \[ 4 \log 3 - 4 \log 2 - \log 5 = \frac{2M}{161} + \frac{2M}{3.161^3} + &c. \] \[ \log 5 - 5 \log 3 + 2 \log 7 = \frac{2M}{244} + \frac{2M}{3.244^3} + &c. \]

Let \( \log 2, \log 3, \log 5, \) and \( \log 7, \) be now considered as four unknown quantities, and by resolving these equations in the usual manner, (see ALGEBRA, Sect. VII.) the logarithms may be determined.

Resuming once more the formula

\[ \log x = 2M \left\{ \frac{x-1}{x+1} + \frac{1}{3} \left( \frac{x-1}{x+1} \right)^3 + &c. \right\} \] Nature of Logarithms, let \( \frac{n^3(n+5)(n-5)}{(n+3)(n-3)(n+4)(n-4)} \) be substituted in it &c.

instead of \( x \), then, by this substitution \( \frac{x-1}{x+1} \) will become

\[ \frac{-72}{n^4-25n^2+72} \]

the formula will be transformed to

\[ \log_{\cdot} \frac{n^3(n+5)(n-5)}{(n+3)(n-3)(n+4)(n-4)} = -2M \left\{ \frac{72}{n^4-25n^2+72} + \frac{72}{(n^4-25n^2+72)^3} + \text{&c.} \right\} \]

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

\[ 2 \log_{\cdot} n + \log_{\cdot}(n+5) + \log_{\cdot}(n-5) - \log_{\cdot}(n+3) - \log_{\cdot}(n-3) - \log_{\cdot}(n+4) - \log_{\cdot}(n-4) + Q = 0 \]

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

When it is required to find the logarithm of a high number, as for example 1231, we may proceed as follows:

\[ \log_{\cdot} 1231 = \log_{\cdot}(1230+1) = \log_{\cdot} \left\{ 1230 \left( 1 + \frac{1}{1230} \right) \right\} = \log_{\cdot} 1230 + \log_{\cdot} \left( 1 + \frac{1}{1230} \right). \]

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

\[ = \log_{\cdot} 120 + \log_{\cdot} \left( 1 + \frac{1}{40} \right) \]

\[ \log_{\cdot} 120 = \log_{\cdot} (2^3 \times 3 \times 5) = 3 \log_{\cdot} 2 + \log_{\cdot} 3 + \log_{\cdot} 5 \]

Therefore

\[ \log_{\cdot} 1231 = 4 \log_{\cdot} 2 + \log_{\cdot} 3 + 2 \log_{\cdot} 5 + \log_{\cdot} \left( 1 + \frac{1}{40} \right) + \log_{\cdot} \left( 1 + \frac{1}{1230} \right) \]

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 h formulas already delivered.

Having now explained, at considerable length, the theory of logarithms upon principles purely analytical, such being, as we conceive, the most natural way of reasoning concerning the properties of number, we shall conclude this section by stating briefly the ground upon which it was referred to the principles of geometry by the mathematicians of the 17th century. Let C (fig. 2.) be the centre, and CH, CK the asymptotes of an hyperbola. In either of these let there be taken any number of continual proportionals CA, CB, CD, CE, &c. then if B b, D d, E e, &c. be drawn parallel to the other asymptote, meeting the curve in a, b, d, e, &c. the hyperbolic spaces A a b B, B b d D, D d e E, &c. are equal to one another; also if straight lines be drawn from C to the points a, b, d, e, &c., the hyperbolic sectors a C b, b C d, d C e, &c. shall also be equal (Conic Sections, Part III. prop. 30.). Now, since it appears by this proposition that the segments CA, CB, CD, CE, &c. of the asymptote being taken in continued geometrical progression, the corresponding hyperbolic areas A a b B, A a d D, A a e E, &c. constitute a series of quantities in continued arithmetical progression, it is evident that the two series will have, in respect to each other, the same properties as numbers and their logarithms; so that, if we assume CA any segment of the asymptote as the representative of unity, and suppose CB, CD, CE, &c. to be the representatives of other numbers, the hyperbolic areas, A a b B, A a d D, A a e E will be the geometrical representatives of the logarithms of these numbers; and so also will the hyperbolic sectors C a b, C b d, C d e, &c.

Let CA (the line denoting unity) be the side of a rhombus CA a L inferred at the vertex of the hyperbola, and let CP = n × CA (n being put for any number); draw PP parallel to CL meeting the hyperbola in P, then it may be shewn, by the methods usually employed in reasoning about curvilinear areas, that the area of the rhombus A a LC is to the hyperbolic area A a P P as 1 to the Napierian logarithm of the number n. Therefore if the hyperbola be equilateral, so that A a Lc is a square, &c. consequently its area = 1 × 1 = 1, the Napierian logarithm of n, and the area A a P P may be taken as the mutual representatives of each other. It is this circumstance which induced mathematicians to call these logarithms hyperbolic. But with equal propriety might the logarithms of any other system be called hyperbolic, as they may be equally expressed by the area of the equilateral hyperbola, or indeed by the area of any hyperbola whatever, (see Fluxions, § 152. Ex. 5.).

SECT. II.

DESCRIPTION AND USE OF THE TABLE.

THE common system of logarithms is so constructed, that, o being the logarithm of unity, or 1, the logarithm of 10 is 1; by which it happens that the logarithm of 100 is 2, that of 1000 is 3, and so on. Also, the logarithm of \( \frac{1}{10} \), or .1, is —1, that is, 1 considered as subtractive; or, in the language of algebra, minus one; and the logarithm of \( \frac{1}{100} \) or .01, is —2; and the logarithm of .001 is —3, and so on, as in the following short table.

<table> <tr> <th>Numbers.</th> <th>Logarithms.</th> </tr> <tr> <td>.001</td> <td>-3</td> </tr> <tr> <td>.00</td> <td>-2</td> </tr> <tr> <td>.1</td> <td>-1</td> </tr> <tr> <td>1</td> <td>0</td> </tr> <tr> <td>10</td> <td>1</td> </tr> <tr> <td>100</td> <td>2</td> </tr> <tr> <td>1000</td> <td>3</td> </tr> <tr> <td>&amp;c.</td> <td>&amp;c.</td> </tr> </table>

As the terms of the geometrical progression 1, 10, 100, &c. continued backwards as well as forward, are the only numbers whose logarithms are integers; the logarithms of all other numbers whatever must be either fractions or mixt numbers. Accordingly, the logarithms of all numbers, whether integer or mixt, between 1 and 10 are expressed by decimal fractions less than Description than unity. The logarithms of numbers between 10 and 100 are expressed by mixt numbers composed of unity and a decimal fraction. The logarithms of numbers between 100 and 1000 are expressed by mixt numbers composed of the number 2 and a decimal fraction, and so on. On the other hand, the logarithm of any vulgar or decimal fraction less than 1, but greater than \( \frac{1}{10} \) or .1, will be some negative decimal fraction between 0 and -1; and the logarithm of any fraction between .1 and .01, will be a negative mixed quantity between -1 and -2, and so on.

But it must be remarked, that any fraction, or mixt number, considered as entirely negative, may always be transformed into another mixt number of equal value, that shall have its integer part negative, but its fractional part positive, by diminishing the integer by unity, and increasing the fractional part by the same quantity. Thus let the mixt quantity be \( -2\frac{3}{10} \), which may be also written thus \( -2\frac{3}{10} \). Let the integer -2 be diminished by 1, and the result is \( -2-1 = -3 \). Also, let the fraction \( \frac{3}{10} \) be increased by 1, and it becomes \( \frac{3}{10} + 1 = \frac{13}{10} \); therefore the fraction \( -2\frac{3}{10} \) or \( -2.3 \), when transformed, is \( -3+\frac{7}{10} \), or \( -3+.7 \), which may be written thus, 3.7; where the negative sign is placed over the integer to indicate that it is the only part of the expression that is considered as negative, the other part, viz. .7, being reckoned positive.

Since therefore any fractional or mixt quantity, considered as entirely negative, is equivalent to another mixt quantity, the integer part of which only is negative, but the fractional part positive, it is evident that instead of expressing the logarithms of fractions by numbers considered as entirely negative, we may express them by numbers having their integer parts negative, and their decimal parts positive; and it is usual so to express them. Thus the logarithm of .03, instead of being expressed by \( -1.52288 \), that is, by \( -1-.52288 \), is usually expressed by 2.47712, by which is to be understood \( -2+.47712 \). Again, the logarithm of .7, which, if considered as entirely negative, would be \( -1.5490 \), is otherwise 1.84510.

As the logarithms of any series of numbers forming a geometrical progression, the common ratio of which is 10, will exceed each other by the logarithm of 10, that is, by 1, it follows that the logarithms of all numbers denoted by the same figures, and differing only in the position of the decimal point, will have the decimal part of their logarithms the same; but the integers standing before the decimals will be different, and will be positive or negative, according as the numbers are whole or fractional, as in these examples.

<table> <tr> <th>Numbers.</th> <th>Logarithms.</th> </tr> <tr> <td>69150</td> <td>4.83980</td> </tr> <tr> <td>6915</td> <td>3.83980</td> </tr> <tr> <td>691.5</td> <td>2.83980</td> </tr> <tr> <td>69.15</td> <td>1.83980</td> </tr> <tr> <td>6.915</td> <td>0.83980</td> </tr> <tr> <td>.6915</td> <td>1.83980</td> </tr> <tr> <td>.06915</td> <td>2.83980</td> </tr> </table>

The integer figure of a logarithm, is called its index or characteristic; and it is always less by one than the number of integer figures which the natural number consists of; or it is equal to the distance of the first figure from the place of units or first place of integers, whether on the left or on the right of it.

The table of logarithms given at the end of this article, contains the decimal parts of the logarithms of all numbers from 1 to 10,000; and indeed of all numbers which can be expressed by four figures, preceded or followed by any numbers of cyphers, such as the numbers 367500, .002795, &c. The index, however, is not put down; but it is easily supplied by the rule which has just now been given. The table also contains the differences of the logarithms of all numbers from 1000 to 10,000, by means of which the logarithm of any number consisting of five figures may be easily obtained.

1. To find the logarithm of any number consisting of four or any smaller number of figures. Look for the number in the columns titled at the top Numbers; and in the same line with it, on the right, in the column of logarithms, will be found the decimal part of its logarithm, to which supply the decimal point, and its index according to rule delivered above. Thus,

<table> <tr> <th>The log. of 9 is found to be 0.95424</th> </tr> <tr> <td>of 17</td> <td>1.23045</td> </tr> <tr> <td>of 2.63</td> <td>0.41996</td> </tr> <tr> <td>of 13.42</td> <td>2.12775</td> </tr> <tr> <td>of 6280</td> <td>3.79796</td> </tr> <tr> <td>of 3749</td> <td>3.57392</td> </tr> <tr> <td>of .6027</td> <td>1.78010</td> </tr> <tr> <td>of .00234</td> <td>3.36922</td> </tr> <tr> <td>of 852600</td> <td>5.93075</td> </tr> </table>

2. To find the logarithm of a number consisting of five figures.

Find the decimal part of the logarithm of the first four figures of the number, (that is, find the logarithm of the proposed number as if the last figure were a cipher), by the preceding rule, and find the difference between that logarithm and the next greater, as given in the column of differences (to the right of the column of logarithms). Then state this proportion:

As 10, To the tabular difference, So is the last, or fifth figure of the number, To a fourth proportional;

which being added to the former logarithm, and the decimal point and index supplied, will be the logarithm sought.

Example. Required the logarithm of 186.47. The decimal part of the logarithm of the first four figures, viz. 1864, is .27045, and the difference opposite to it in the column marked D on the top is 23. Therefore we have this proportion:

\[ 10 : 23 :: 7 : \frac{7 \times 23}{10} = 16.1 \]

The fourth proportional is 16.1, or, rejecting the decimal part, .16 nearly; therefore,

\[ \text{to log. of 1684} \quad .27045 \\ \text{add} \qquad \frac{16}{16} \\ \text{the log. of 168.47 is} \quad 2.27061 \]

3. To find the logarithm of a vulgar fraction or mixt number.

Either reduce the vulgar fraction to a decimal, and find its logarithm as above, or else (having reduced the mixt number to an improper fraction) subtract the logarithm of the denominator from the logarithm of the numerator, and the remainder will be the logarithm of the fraction sought.

Ex. 1. To find the logarithm of \( \frac{7}{8} \).

From the log. of 3 \hspace{2cm} 0.47712 Subtract the log. of 16 \hspace{2cm} 1.20412

Rem. log. of \( \frac{7}{8} \) or of .1875 \hspace{2cm} 1.27300

Here, as the lower number is greater than the upper, the remainder must be negative; the subtraction, however, is so performed, that the decimal part of the remainder is positive, and the integer negative.

Ex. 2. To find the logarithm of \( 13\frac{1}{4} \) or \( \frac{55}{4} \).

From log. of 55 \hspace{2cm} 1.74036 Subtract log. of 4 \hspace{2cm} 0.60206

Rem. log. of \( 13\frac{1}{4} \) or of 13.75 \hspace{2cm} 1.13830

4. To find the number corresponding to any given logarithm.

Seek the decimal part of the proposed logarithm in the column of logarithms, and if it be found exactly, the figures of the number corresponding to it will be found in the same line with it in the column of numbers. If the index of the given logarithm is 3, the four figures of the numbers thus found are integers; but if it be 2, the three first figures are integers, and the fourth is a decimal, and so on; the number of integer figures before the decimal point being always one greater than the index, if it be positive; but if it be negative, the number sought will be a decimal, and the number of cyphers between the decimal point and first significant figure will be one less than the index.—Examples. The number corresponding to the logarithm 3.57392 is 3749. The number corresponding to 1.12775 is 13.42. The number corresponding to 3.36022 is .00234, and so on.

But if the given logarithm is not exactly found in the table, subtract the next less tabular logarithm from it, and take the difference between that logarithm, and the next greater (as given in the column of differences). Then state this proportion:

As the difference, taken from the table, Is to 10, So is the difference between the given logarithm and the next less, To a fourth proportional,

which being annexed to the four figures corresponding to the logarithm next less than the given one, will be the logarithm required.

Example. Find the number answering to the logarithm 4.13278.

The dec. part of given log. is .13278 That of next less, viz. log. of 1357, is .13258 Difference 20

The tabular difference is 32, therefore we have this proportion,

\[ 32 : 10 :: 20 : \frac{20 \times 10}{32} = 6 \text{ nearly}. \]

Therefore the number corresponding to the proposed logarithm is .13576.

In like manner may the numbers to the following logarithms be found.

<table> <tr> <th>Logarithms.</th> <th>Numbers.</th> </tr> <tr> <td>1.23457</td> <td>17.162</td> </tr> <tr> <td>3.73430</td> <td>5423.8</td> </tr> <tr> <td>1.09214</td> <td>12363</td> </tr> <tr> <td>4.61230</td> <td>49954</td> </tr> </table>

The table of logarithms of numbers is followed by a Table of logarithmic Sines and Tangents, for every minute of the quadrant, with their differences. For the explanation of this table we refer to Trigonometry, to which branch of mathematics it is intended to be applied.

We shall now give practical rules, illustrated by examples, for performing the different operations of arithmetic by logarithms.

MULTIPLICATION BY LOGARITHMS.

RULE.

Take out the logarithms of the factors from the table; then add them together, and their sum will be the logarithm of the product required. Then find, by inspection of the table, the natural number answering to their sum, and it will be the product required.

Observing to add what is to be carried from the decimal part of the logarithm to the positive index or indices, or else subtract it from the negative.

Also adding the indices together when they are of the same kind, that is, both positive or both negative; but subtracting the less from the greater when the one is positive and the other negative, and prefixing the sign of the greater to the remainder.

EXAMPLES.

Ex. 1. To multiply 2.314 by 50.62.

<table> <tr> <th>Numbers.</th> <th>Logarithms.</th> </tr> <tr> <td>2.314</td> <td>0.36436</td> </tr> <tr> <td>50.62</td> <td>1.70432</td> </tr> <tr> <td colspan="2">Product 117.13</td> </tr> <tr> <td colspan="2">2.06868</td> </tr> </table>

Ex. 2. To multiply 2.5819 by 3.4573.

<table> <tr> <th>Numbers.</th> <th>Logarithms.</th> </tr> <tr> <td>2.5819</td> <td>0.41194</td> </tr> <tr> <td>3.4573</td> <td>0.53874</td> </tr> <tr> <td colspan="2">Prod. 8.9265</td> </tr> <tr> <td colspan="2">0.95068</td> </tr> </table>

Ex. 3. To multiply 39.02, and 397.16, and .03147 together.

<table> <tr> <th>Numbers.</th> <th>Logarithms.</th> </tr> <tr> <td>39.02</td> <td>1.59129</td> </tr> <tr> <td>397.16</td> <td>2.77609</td> </tr> <tr> <td>.03147</td> <td>2.49790</td> </tr> <tr> <td colspan="2">Prod. 753.3 2.86528</td> </tr> </table>

Here the sum of the positive indices, together with 1 which we carry, is 4, and from this we subtract 2, because of the negative index —2.

Ex. 4. To multiply 3.586 and 2.1046, and 0.8372 and 0.0294 all together.

<table> <tr> <th>Numbers.</th> <th>Logarithms.</th> </tr> <tr> <td>3.586</td> <td>0.55461</td> </tr> <tr> <td>2.1046</td> <td>0.32317</td> </tr> <tr> <td>0.8372</td> <td>1.92283</td> </tr> <tr> <td>0.0294</td> <td>2.46835</td> </tr> <tr> <td colspan="2">1.26896</td> </tr> </table>

Here the 2 to carry cancels the —2, and there remains the —1 to set down.

DIVISION by LOGARITHMS.

RULE.

SUBTRACT the logarithm of the divisor from the logarithm of the dividend, and the number answering to the remainder will be the logarithm of the quotient required.

Observing to change the sign of the index of the divisor from positive to negative, or from negative to positive; then take the sum of the indices if they be of the same name, or their difference when they have different signs, with the sign of the greater for the index to the logarithm of the quotient.

Alfo, when 1 is borrowed in the left-hand place of the decimal part of the logarithm, add it to the index of the divisor when that index is positive, but subtract it when negative; then let the index arising from thence be changed, and work with it as before.

EXAMPLES.

Ex. To divide 24163 by 4567.

<table> <tr> <th>Numbers.</th> <th>Logarithms.</th> </tr> <tr> <td>Divid. 24163</td> <td>4.38315</td> </tr> <tr> <td>Divif. 4567</td> <td>3.65963</td> </tr> <tr> <td>Quot. 5.2928</td> <td>0.72352</td> </tr> </table>

Ex. To divide 37.15 by 523.76.

<table> <tr> <th>Numbers.</th> <th>Logarithms.</th> </tr> <tr> <td>Divid. 37.15</td> <td>1.56996</td> </tr> <tr> <td>Divif. 523.76</td> <td>2.71913</td> </tr> <tr> <td>Quot. .07093</td> <td>2.85083</td> </tr> </table>

Ex. 3. Divide .06314 by .007241.

<table> <tr> <th>Number.</th> <th>Logarithms.</th> </tr> <tr> <td>Divid. .06314</td> <td>2.80030</td> </tr> <tr> <td>Divif. .007241</td> <td>3.85980</td> </tr> <tr> <td>Quot. 8.720</td> <td>0.94050</td> </tr> </table>

Here 1 carried from the decimals to the —3 makes it —2, which taken from the other —2, leaves 0 remaining.

Ex. 4. Divide .7438 by 12.947.

<table> <tr> <th>Numbers.</th> <th>Logarithms.</th> </tr> <tr> <td>Divid. .7438</td> <td>1.87146</td> </tr> <tr> <td>Divif. 12.947</td> <td>1.11218</td> </tr> <tr> <td>Quot. .057449</td> <td>2.75928</td> </tr> </table>

Here the 1 taken from the —1 makes it become —2 to set down.

PROPORTION BY LOGARITHMS.

RULE.

ADD the logarithms of the second and third terms, and from the sum subtract the logarithm of the first term by the foregoing rules, the remainder will be the logarithm of the fourth term required.

Or in any compound proportion whatever, add together the logarithms of all the terms that are to be multiplied; and from that sum take the sum of the others, the remainder will be the logarithm of the anfwer.

But, instead of subtracting any logarithm, we may add its arithmetical complement, and the result will be the fame. By the arithmetical complement is meant the logarithm of the reciprocal of the given number, or the remainder by taking the given logarithm from 0, or from 10, changing the beginning of the scale from 0 to 10; the eafieft way of doing which is to begin at the left hand, and subtract each figure from 9, except at the last significant figure on the right hand, which must be subtracted from 10. But when the index is negative, it must be added to 9, and the rest subtracted as before; and for every complement that is added, subtract 10 from the last sum of the indices.

EXAMPLES.

Ex. 1. Find a fourth proportional to 72.34, 2.519, and 357.48.

<table> <tr> <th>Numbers.</th> <th>Logarithms.</th> </tr> <tr> <td>As 72.34</td> <td>1.85938</td> </tr> <tr> <td>To 2.519</td> <td>0.40123</td> </tr> <tr> <td>So is 357.48</td> <td>2.55325</td> </tr> <tr> <td colspan="2">2.95448</td> </tr> <tr> <td>To 12.448</td> <td>1.09510</td> </tr> </table>

Here the logarithms of the second and third terms are added together, and the logarithm of the first term is subtracted from the sum; but by taking the arithmeti-

Description and Use of the Table.

cal complement of the first term, the work might stand thus:

As 72.34 Comp. log. 8.14062 To 2.519 0.40123 So is 357.48 2.55325

To 12.448 1.09510

Ex. 2. If the interest of 100l. for a year, or 365 days, be 4.5, What will be the interest of 279.25l. for 274 days?

As { 100 Comp. long. { 8.00000 { 365 { 7.43771 To { 279.25 { 2.44599 { 274 { 2.43775 So is 4.5 { 0.65321

To 9.4333 0.97466

Here, instead of subtracting the sum of the logarithms of 100 and 365, we add the arithmetical complement of the logarithms of these numbers, and subtract 20 from the sum of the indices.

INVOLUTION BY LOGARITHMS.

RULE.

Multiply the logarithm of the given number by the index of the power, and the number answering to the product will be the power required.

Note.—In multiplying a logarithm with a negative index by a positive number, the product will be negative. But what is to be carried from the decimal part of the logarithm will always be positive. And therefore the difference will be the index of the product, and is always to be made of the same kind with the greater.

EXAMPLES.

Ex. 1. To square the number 2.579.

<table> <tr> <th>Number.</th> <th>Logarithm.</th> </tr> <tr> <td>Root 2.569</td> <td>0.41145</td> </tr> <tr> <td>The index 2</td> <td></td> </tr> <tr> <td>Power 6.6513</td> <td>0.82290</td> </tr> </table>

Ex. 2. To find the cube of 3.0715.

<table> <tr> <th>Number.</th> <th>Logarithm.</th> </tr> <tr> <td>Root 3.0715</td> <td>0.48735</td> </tr> <tr> <td>The index 3</td> <td></td> </tr> <tr> <td>Power 28.976</td> <td>1.46205</td> </tr> </table>

Ex. 3. To raise .09163 to the fourth power.

<table> <tr> <th>Number.</th> <th>Logarithm.</th> </tr> <tr> <td>Root .09163</td> <td>2.96204</td> </tr> <tr> <td>The index 4</td> <td></td> </tr> <tr> <td>Power .000070495</td> <td>5.84816</td> </tr> </table>

Here 4 times the negative index being —8, and 3 to carry, the difference —5, is the index of the product.

Ex. 4. To raise 1.0045 to the 365th power.

<table> <tr> <th>Number.</th> <th>Logarithm.</th> </tr> <tr> <td>Root 1.0045</td> <td>0.00195</td> </tr> <tr> <td>The index 365</td> <td>395</td> </tr> <tr> <td></td> <td>975</td> </tr> <tr> <td></td> <td>1170</td> </tr> <tr> <td></td> <td>585</td> </tr> <tr> <td>Power 5.1493</td> <td>.71175</td> </tr> </table>

EVOLUTION BY LOGARITHMS.

RULE.

Divide the logarithm of the number by the index of the root, and the number answering to the quotient is the root sought.

When the index of the logarithm to be divided is negative, and does not exactly contain the divisor without some remainder, increase the index by such a number as will make it exactly divisible by the index of the root, carrying the units borrowed as so many tens to the left-hand place of the decimal, and then divide as in whole numbers.

EXAMPLES.

Ex. 1. Find the square root of 2.

<table> <tr> <th>Number.</th> <th>Logarithm.</th> </tr> <tr> <td>Power .2</td> <td>2)0.30103</td> </tr> <tr> <td>Root 1.4142</td> <td>0.15051</td> </tr> </table>

Ex. 2. Find the 10th root of 365.

<table> <tr> <th>Number.</th> <th>Logarithm.</th> </tr> <tr> <td>Power 365</td> <td>10)2.56229</td> </tr> <tr> <td>Root 1.804</td> <td>0.25623</td> </tr> </table>

Ex. 3. To find \( \sqrt{.093} \).

<table> <tr> <th>Number.</th> <th>Logarithm.</th> </tr> <tr> <td>Power .093</td> <td>2)2.96848</td> </tr> <tr> <td>Root .30496</td> <td>1.48424</td> </tr> </table>

Here the divisor 2 is contained exactly in the negative index —2, and therefore the index of the quotient is —1.

Ex. 4. To find \( 3\sqrt{.00048} \).

<table> <tr> <th>Number.</th> <th>Logarithm.</th> </tr> <tr> <td>Power .00048</td> <td>3)4.68124</td> </tr> <tr> <td>Root .078298</td> <td>2.89375</td> </tr> </table>

Here the divisor 3, not being exactly contained in —4, it is augmented by 2 to make up 6, in which the divisor is contained just 2 times, then the 2 thus borrowed being carried to the decimal figure 6, makes 26, which divided by 3 gives 8, &c. <table> <tr> <th>N.</th><th>Log.</th><th>N.</th><th>Log.</th><th>N.</th><th>Log.</th><th>N.</th><th>Log.</th><th>N.</th><th>Log.</th><th>N.</th><th>Log.</th><th>N.</th><th>Log.</th> </tr> <tr><td>1</td><td>0.0000</td><td>6377815</td><td>1207918</td><td>18025527</td><td>24038221</td><td>30047712</td><td>36055630</td><td>42062325</td></tr> <tr><td>2</td><td>0.0103</td><td>6178533</td><td>12108279</td><td>18125768</td><td>24138202</td><td>30147957</td><td>30155751</td><td>42162428</td></tr> <tr><td>3</td><td>0.0471</td><td>6279239</td><td>12208636</td><td>18226007</td><td>24238482</td><td>30248001</td><td>30255871</td><td>42262531</td></tr> <tr><td>4</td><td>0.0771</td><td>6379934</td><td>12308991</td><td>18326245</td><td>24338661</td><td>30348144</td><td>30355991</td><td>42362634</td></tr> <tr><td>5</td><td>0.1026</td><td>6480618</td><td>12409342</td><td>18426482</td><td>24438739</td><td>30448287</td><td>30456110</td><td>42462737</td></tr> <tr><td>6</td><td>0.1297</td><td>6581291</td><td>12509691</td><td>18526717</td><td>24538917</td><td>30548430</td><td>30556229</td><td>42562839</td></tr> <tr><td>7</td><td>0.1581</td><td>6681954</td><td>12610037</td><td>18626951</td><td>24639094</td><td>30648572</td><td>30656348</td><td>42662941</td></tr> <tr><td>8</td><td>0.1850</td><td>6782607</td><td>12710380</td><td>18727184</td><td>24739270</td><td>30748714</td><td>30756467</td><td>42763043</td></tr> <tr><td>9</td><td>0.2109</td><td>6883251</td><td>12810721</td><td>18827416</td><td>24839445</td><td>30848855</td><td>30856585</td><td>42863144</td></tr> <tr><td>10</td><td>0.2354</td><td>6983885</td><td>12911059</td><td>18927646</td><td>24939620</td><td>30948996</td><td>30956703</td><td>42963246</td></tr> <tr><td>11</td><td>0.2600</td><td>7084510</td><td>13011394</td><td>19027875</td><td>25039794</td><td>31049136</td><td>31056820</td><td>43063347</td></tr> <tr><td>12</td><td>0.2849</td><td>7185126</td><td>13111727</td><td>19128103</td><td>25139967</td><td>31149276</td><td>31156937</td><td>43163448</td></tr> <tr><td>13</td><td>0.3091</td><td>7285733</td><td>13212057</td><td>19228332</td><td>25240140</td><td>31249415</td><td>31257054</td><td>43263548</td></tr> <tr><td>14</td><td>0.3334</td><td>7386332</td><td>13312385</td><td>19328563</td><td>25340312</td><td>31349554</td><td>31357171</td><td>43363649</td></tr> <tr><td>15</td><td>0.3575</td><td>7486923</td><td>13412710</td><td>19428780</td><td>25440483</td><td>31449693</td><td>31457287</td><td>43463749</td></tr> <tr><td>16</td><td>0.3816</td><td>7587506</td><td>13513033</td><td>19529003</td><td>25540654</td><td>31549831</td><td>31557403</td><td>43563849</td></tr> <tr><td>17</td><td>0.4057</td><td>7688081</td><td>13613354</td><td>19629226</td><td>25640824</td><td>31649969</td><td>31657519</td><td>43663949</td></tr> <tr><td>18</td><td>0.4298</td><td>7788649</td><td>13713672</td><td>19729447</td><td>25740993</td><td>31750106</td><td>31757636</td><td>43764048</td></tr> <tr><td>19</td><td>0.4539</td><td>7889209</td><td>13813988</td><td>19829667</td><td>25841162</td><td>31850243</td><td>31857749</td><td>43864147</td></tr> <tr><td>20</td><td>0.4779</td><td>7989763</td><td>13914311</td><td>19929885</td><td>25941330</td><td>31950379</td><td>31957864</td><td>43964246</td></tr> <tr><td>21</td><td>0.5019</td><td>8090319</td><td>14014613</td><td>20031013</td><td>26041497</td><td>32050513</td><td>32057978</td><td>44064345</td></tr> <tr><td>22</td><td>0.5259</td><td>8190874</td><td>14114922</td><td>20132302</td><td>26141664</td><td>32150651</td><td>32158092</td><td>44164444</td></tr> <tr><td>23</td><td>0.5499</td><td>8291421</td><td>14215229</td><td>20232553</td><td>26241830</td><td>32250786</td><td>32258206</td><td>44264542</td></tr> <tr><td>24</td><td>0.5739</td><td>8391968</td><td>14315534</td><td>20332750</td><td>26341996</td><td>32350920</td><td>32358320</td><td>44364640</td></tr> <tr><td>25</td><td>0.5979</td><td>8492428</td><td>14415836</td><td>20432963</td><td>26442160</td><td>32451055</td><td>32458433</td><td>44464738</td></tr> <tr><td>26</td><td>0.6218</td><td>8592984</td><td>14516137</td><td>20533173</td><td>26542325</td><td>32551186</td><td>32558546</td><td>44564836</td></tr> <tr><td>27</td><td>0.6458</td><td>8693450</td><td>14616435</td><td>20633387</td><td>26642488</td><td>32651322</td><td>32658659</td><td>44664933</td></tr> <tr><td>28</td><td>0.6697</td><td>8793916</td><td>14716732</td><td>20733597</td><td>26742651</td><td>32751455</td><td>32758771</td><td>44765031</td></tr> <tr><td>29</td><td>0.6937</td><td>8894382</td><td>14817020</td><td>20833806</td><td>26842813</td><td>32851587</td><td>32858883</td><td>44865128</td></tr> <tr><td>30</td><td>0.7176</td><td>8994849</td><td>14917319</td><td>20934015</td><td>26942975</td><td>32951720</td><td>32958995</td><td>44965225</td></tr> <tr><td>31</td><td>0.7416</td><td>9095316</td><td>15017609</td><td>21034222</td><td>27043136</td><td>33051851</td><td>33059106</td><td>45065321</td></tr> <tr><td>32</td><td>0.7655</td><td>9195784</td><td>15117898</td><td>21134328</td><td>27143297</td><td>33151983</td><td>33159218</td><td>45165418</td></tr> <tr><td>33</td><td>0.7895</td><td>9296252</td><td>15218184</td><td>21234534</td><td>27243457</td><td>33252114</td><td>33259329</td><td>45265514</td></tr> <tr><td>34</td><td>0.8134</td><td>9396719</td><td>15318466</td><td>21334638</td><td>27343616</td><td>33352244</td><td>33359439</td><td>45365610</td></tr> <tr><td>35</td><td>0.8374</td><td>9497187</td><td>15418752</td><td>21434841</td><td>27443775</td><td>33452373</td><td>33459550</td><td>45465706</td></tr> <tr><td>36</td><td>0.8613</td><td>9597754</td><td>15519033</td><td>21534944</td><td>27543933</td><td>33552504</td><td>33559660</td><td>45565801</td></tr> <tr><td>37</td><td>0.8853</td><td>9698222</td><td>15619312</td><td>21635145</td><td>27644091</td><td>33652634</td><td>33659775</td><td>45665906</td></tr> <tr><td>38</td><td>0.9092</td><td>9798687</td><td>15719590</td><td>21735346</td><td>27744248</td><td>33752763</td><td>33759889</td><td>45766011</td></tr> <tr><td>39</td><td>0.9332</td><td>9899153</td><td>15819866</td><td>21835448</td><td>27844404</td><td>33852892</td><td>33859998</td><td>45866116</td></tr> <tr><td>40</td><td>0.9571</td><td>9999620</td><td>15920140</td><td>21935644</td><td>27944560</td><td>33953021</td><td>33956007</td><td>45966216</td></tr> <tr><td>41</td><td>0.9811</td><td>10000000</td><td>16020412</td><td>22035816</td><td>28044716</td><td>34053148</td><td>34056226</td><td>46066321</td></tr> <tr><td>42</td><td>1.0051</td><td>10100432</td><td>16120683</td><td>22134349</td><td>28144871</td><td>34153275</td><td>34156343</td><td>46166426</td></tr> <tr><td>43</td><td>1.0291</td><td>10200860</td><td>16220952</td><td>22234035</td><td>28245025</td><td>34253403</td><td>34256459</td><td>46266531</td></tr> <tr><td>44</td><td>1.0531</td><td>10301284</td><td>16321219</td><td>22334380</td><td>28345179</td><td>34353529</td><td>34356575</td><td>46366636</td></tr> <tr><td>45</td><td>1.0771</td><td>10401703</td><td>16421486</td><td>22434525</td><td>28445332</td><td>34453656</td><td>34456692</td><td>46466741</td></tr> <tr><td>46</td><td>1.1011</td><td>10502119</td><td>16521745</td><td>22534718</td><td>28545484</td><td>34553782</td><td>34556808</td><td>46566847</td></tr> <tr><td>47</td><td>1.1251</td><td>10602531</td><td>16622011</td><td>22635411</td><td>28645637</td><td>34653908</td><td>34656924</td><td>46666953</td></tr> <tr><td>48</td><td>1.1491</td><td>10702938</td><td>16722272</td><td>22735603</td><td>28745788</td><td>34754033</td><td>34757040</td><td>46767059</td></tr> <tr><td>49</td><td>1.1731</td><td>10803342</td><td>16822531</td><td>22835793</td><td>28845939</td><td>34854158</td><td>34857156</td><td>46867165</td></tr> <tr><td>50</td><td>1.1971</td><td>10903743</td><td>16922789</td><td>22935984</td><td>28946066</td><td>34954283</td><td>34957272</td><td>46967271</td></tr> <tr><td>51</td><td>1.2211</td><td>11004139</td><td>17023043</td><td>23036173</td><td>29046240</td><td>35054407</td><td>35057388</td><td>47067377</td></tr> <tr><td>52</td><td>1.2451</td><td>11104532</td><td>17123300</td><td>23136361</td><td>29146389</td><td>35154533</td><td>35157504</td><td>47167483</td></tr> <tr><td>53</td><td>1.2691</td><td>11204922</td><td>17223553</td><td>23236549</td><td>29246538</td><td>35254654</td><td>35257620</td><td>47267599</td></tr> <tr><td>54</td><td>1.2931</td><td>11305308</td><td>17323803</td><td>23336736</td><td>29346687</td><td>35354777</td><td>35357736</td><td>47367715</td></tr> <tr><td>55</td><td>1.3171</td><td>11405690</td><td>17424055</td><td>23436922</td><td>29446835</td><td>35454900</td><td>35457852</td><td>47467831</td></tr> <tr><td>56</td><td>1.3411</td><td>11506070</td><td>17524304</td><td>23537107</td><td>29546982</td><td>35555023</td><td>35557968</td><td>47567947</td></tr> <tr><td>57</td><td>1.3651</td><td>11606446</td><td>17624551</td><td>23637291</td><td>29647129</td><td>35655145</td><td>35658084</td><td>47667761</td></tr> <tr><td>58</td><td>1.3891</td><td>11706819</td><td>17724797</td><td>23737475</td><td>29747276</td><td>35755267</td><td>35758199</td><td>47767877</td></tr> <tr><td>59</td><td>1.4131</td><td>11807188</td><td>17825042</td><td>23837658</td><td>29847422</td><td>35855388</td><td>35858315</td><td>47867993</td></tr> <tr><td>60</td><td>1.4371</td><td>11907555</td><td>17925283</td><td>23937809</td><td>29947567</td><td>35955509</td><td>35958431</td><td>47968034</td></tr> <tr><td>61</td><td>1.4611</td><td>12007918</td><td>18025527</td><td>24038021</td><td>30047712</td><td>36055630</td><td>36058547</td><td>48068124</td></tr> </table> <table> <tr> <th>N.</th><th>Log.</th> <th>N.</th><th>Log.</th> <th>N.</th><th>Log.</th> <th>N.</th><th>Log.</th> <th>N.</th><th>Log.</th> <th>N.</th><th>Log.</th> <th>N.</th><th>Log.</th> </tr> <tr> <td>480</td><td>68124</td> <td>540</td><td>73359</td> <td>600</td><td>78155</td> <td>660</td><td>81954</td> <td>720</td><td>87333</td> <td>780</td><td>89229</td> <td>840</td><td>92428</td> </tr> <tr> <td>481</td><td>68215</td> <td>541</td><td>73302</td> <td>601</td><td>77877</td> <td>661</td><td>82002</td> <td>721</td><td>87794</td> <td>781</td><td>89215</td> <td>841</td><td>92496</td> </tr> <tr> <td>482</td><td>68305</td> <td>542</td><td>73400</td> <td>602</td><td>77969</td> <td>662</td><td>82086</td> <td>722</td><td>88254</td> <td>782</td><td>89371</td> <td>842</td><td>92531</td> </tr> <!-- Table continues in this format... --> </table> <table> <tr> <th>N.</th> <th>Log.</th> <th>D.</th> <th>N.</th> <th>Log.</th> <th>D.</th> <th>N.</th> <th>Log.</th> <th>D.</th> <th>N.</th> <th>Log.</th> <th>D.</th> <th>N.</th> <th>Log.</th> <th>D.</th> <th>N.</th> <th>Log.</th> <th>D.</th> <th>N.</th> <th>Log.</th> <th>D.</th> <th>N.</th> <th>Log.</th> <th>D.</th> <th>N.</th> <th>Log.</th> <th>D.</th> <th>N.</th> <th>Log.</th> <th>D.</th> <th>N.</th> <th>Log.</th> <th>D.</th> <th>N.</th> <th>Log.</th> <th>D.</th> <th>N.</th> <th>Log.</th> <th>D.</th> <th>N.</th> <th>Log.</th> <th>D.</th> <th>N.</th> <th>Log.</th> <th>D.</th> <th>N.</th> <th>Log.</th> <th>D.</th> <th>N.</th> <th>Log.</th> <th>D.</th> <th>N.</th> <th>Log.</th> <th>D.</th> <th>N.</th> <th>Log.</th> <th>D.</th> <th>N.</th> <th>Log.</th> <th>D.</th> <th>N.</th> <th>Log.</th> <th>D.</th> <th>N.</th> <th>Log.</th> <th>D.</th> <th>N.</th> <th>Log.</th> <th>D.</th> <th>N.</th> <th>Log.</th> <th>D.</th> <th>N.</th> <th>Log.</th> <th>D.</th> <th>N.</th> <th>Log.</th> <th>D.</th> </tr> <tr><td>96098214</td><td>7.9662035</td><td>42</td><td>102210653</td><td>8.0209720</td><td>43</td><td>108009336</td><td>8.0347893</td><td>41</td><td>114015695</td><td>8.1015093</td><td>39</td><td>120010791</td><td>8.1020754</td><td>36</td><td>126010037</td><td>8.1169965</td><td>33</td><td>132012057</td><td>8.1373490</td><td>35</td><td>138013978</td><td>8.1367731</td><td>37</td></tr> <tr><td>96192872</td><td>7.9687502</td><td>43</td><td>102209923</td><td>8.0204563</td><td>43</td><td>108113383</td><td>8.0413929</td><td>40</td><td>114105726</td><td>8.1105474</td><td>38</td><td>120210799</td><td>8.1200799</td><td>37</td><td>126110072</td><td>8.1176456</td><td>34</td><td>132112105</td><td>8.1361335</td><td>35</td><td>138114019</td><td>8.1352891</td><td>37</td></tr> <tr><td>96283618</td><td>7.9698254</td><td>43</td><td>102209043</td><td>8.0206557</td><td>43</td><td>108310343</td><td>8.0482967</td><td>40</td><td>114205676</td><td>8.1206756</td><td>38</td><td>120310827</td><td>8.1205807</td><td>37</td><td>126210104</td><td>8.1176456</td><td>34</td><td>132212219</td><td>8.1361234</td><td>35</td><td>138214051</td><td>8.1347853</td><td>37</td></tr> <tr><td>96389836</td><td>7.9708412</td><td>43</td><td>102300883</td><td>8.0245579</td><td>42</td><td>108310433</td><td>8.0483861</td><td>40</td><td>114305890</td><td>8.1244851</td><td>38</td><td>120410863</td><td>8.1208563</td><td>37</td><td>126310140</td><td>8.1183424</td><td>34</td><td>132312155</td><td>8.1327565</td><td>33</td><td>138314082</td><td>8.1344842</td><td>37</td></tr> <tr><td>96480438</td><td>7.9718542</td><td>43</td><td>102401030</td><td>8.0253573</td><td>42</td><td>108510353</td><td>8.0575357</td><td>40</td><td>114405885</td><td>8.1244851</td><td>38</td><td>120510809</td><td>8.1208563</td><td>37</td><td>126410175</td><td>8.1186229</td><td>34</td><td>132412168</td><td>8.1324424</td><td>33</td><td>138414114</td><td>8.1321649</td><td>37</td></tr> <tr><td>96584533</td><td>7.9728576</td><td>43</td><td>102501077</td><td>8.0273543</td><td>42</td><td>108510353</td><td>8.0575357</td><td>40</td><td>114505885</td><td>8.1244851</td><td>38</td><td>120609135</td><td>8.1208171</td><td>37</td><td>126510243</td><td>8.1186229</td><td>34</td><td>132512222</td><td>8.1321451</td><td>32</td><td>138514145</td><td>8.1341716</td><td>37</td></tr> <!-- Table continues with more rows... --> </table> <table> <tr> <th style="width:30px">N.</th> <th style="width:30px">Log.</th> <th colspan="2">...</th> <th>N.</th> <th>Log.</th> <th colspan="2">...</th> <th>N.</th> <th>Log.</th> <th colspan="2">...</th> <th>N.</th> <th>Log.</th> <th colspan="2">...</th> <th>N.</th> <th>Log.</th> </tr> <tr> <td>1441</td> <td>15866</td> <td></td> <td></td> <td>1550</td> <td>1791</td> <td></td> <td></td> <td>1560</td> <td>19317</td> <td></td> <td></td> <td>1570</td> <td>19627</td> <td></td> <td></td> <td>1580</td> <td>19951</td> <td></td> <td></td> <td>1600</td> <td>20306</td> </tr> <tr> <td>1442</td> <td>15897</td> <td></td> <td></td> <td>1551</td> <td>17698</td> <td></td> <td></td> <td>1561</td> <td>19358</td> <td></td> <td></td> <td>1571</td> <td>19636</td> <td></td> <td></td> <td>1581</td> <td>19957</td> <td></td> <td></td> <td>1601</td> <td>20318</td> </tr> <!-- ... continue the table --> </table> <table> <tr> <th>N.</th> <th>Log.</th> <th>D.</th> <th>N.</th> <th>Log.</th> <th>D.</th> <th>N.</th> <th>Log.</th> <th>D.</th> <th>N.</th> <th>Log.</th> <th>D.</th> <th>N.</th> <th>Log.</th> <th>D.</th> <th>N.</th> <th>Log.</th> <th>D.</th> <th>N.</th> <th>Log.</th> <th>D.</th> <th>N.</th> <th>Log.</th> <th>D.</th> </tr> <tr><td>1920</td><td>28335</td><td>2</td><td>1980</td><td>29667</td><td>2</td><td>2040</td><td>30963</td><td>2</td><td>2100</td><td>32222</td><td>2</td><td>2160</td><td>33445</td><td>2</td><td>2220</td><td>34635</td><td>2</td><td>2280</td><td>35793</td><td>2</td><td>2340</td><td>36922</td><td>18</td></tr> <tr><td>1921</td><td>28353</td><td>2</td><td>1981</td><td>29688</td><td>2</td><td>2041</td><td>30982</td><td>2</td><td>2101</td><td>32243</td><td>2</td><td>2161</td><td>33456</td><td>2</td><td>2221</td><td>34655</td><td>2</td><td>2281</td><td>35813</td><td>2</td><td>2341</td><td>36940</td><td>19</td></tr> <tr><td>1922</td><td>28375</td><td>2</td><td>1982</td><td>29710</td><td>2</td><td>2042</td><td>31000</td><td>2</td><td>2102</td><td>32263</td><td>2</td><td>2162</td><td>33468</td><td>2</td><td>2222</td><td>34674</td><td>2</td><td>2282</td><td>35832</td><td>2</td><td>2342</td><td>36959</td><td>19</td></tr> <tr><td>1923</td><td>28398</td><td>2</td><td>1983</td><td>29732</td><td>2</td><td>2043</td><td>31021</td><td>2</td><td>2103</td><td>32284</td><td>2</td><td>2163</td><td>33350</td><td>2</td><td>2223</td><td>34694</td><td>2</td><td>2283</td><td>35851</td><td>2</td><td>2343</td><td>36977</td><td>19</td></tr> <tr><td>1924</td><td>28421</td><td>2</td><td>1984</td><td>29754</td><td>2</td><td>2044</td><td>31048</td><td>2</td><td>2104</td><td>32305</td><td>2</td><td>2164</td><td>33352</td><td>2</td><td>2224</td><td>34713</td><td>2</td><td>2284</td><td>35870</td><td>2</td><td>2344</td><td>36996</td><td>19</td></tr> <tr><td>1925</td><td>28443</td><td>2</td><td>1985</td><td>29777</td><td>2</td><td>2045</td><td>31069</td><td>2</td><td>2105</td><td>32325</td><td>2</td><td>2165</td><td>33354</td><td>2</td><td>2225</td><td>34733</td><td>2</td><td>2285</td><td>35889</td><td>2</td><td>2345</td><td>37014</td><td>18</td></tr> <!-- Table continues for all numbers up to 2400 --> </table>

VOL. XII. Part I. <table> <tr> <th>N.</th> <th>Log.</th> <th>D.</th> <th>N.</th> <th>Log.</th> <th>D.</th> <th>N.</th> <th>Log.</th> <th>D.</th> <th>N.</th> <th>Log.</th> <th>D.</th> <th>N.</th> <th>Log.</th> <th>D.</th> </tr> <!-- Table body omitted for brevity --> </table>