1.—HISTORY OF LOGARITHMS.

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

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

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

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

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

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

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

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

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

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

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

If we suppose the number of interpolated means to be very great, it will follow that among the terms of the resulting geometrical series, some one or other will be found nearly equal to any proposed number whatever. Therefore, although the preceding table exhibits the logarithms of 1, 2, 4, 8, 16, &c. but does not contain those of the intermediate numbers, 3, 5, 6, 7, 9, 10, &c. yet it is easy to conceive that a table might be formed by interpolation which should contain, amongst the terms of the geometrical series, all numbers whatever to a certain extent (or at least others very nearly equal to them), together with their logarithms. If such a table were constructed, or at least if such terms History. of the geometrical progression were found, together with their logarithms, as were either accurately equal to, or coincided nearly with, all numbers within certain limits (for example, between 1 and 100,000), then, as often as we had occasion to multiply or divide any numbers contained in that table, we might evidently obtain the products or quotients by the simple operations of addition and subtraction.

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

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

According to this mode of conceiving the line to decrease, it is easy to show that at the end of any successive equal intervals of time from the beginning of the motion, the portions of the line which remain will constitute a decreasing geometrical progression.

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

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

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

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

In the same year, 1624, the celebrated John Kepler published, at Marburg, logarithms of nearly the same kind, under the title of Chilias Logarithmorum ad totidem Numeros Rotundos, premissa Demonstratio legitima Ortu Logarithmorum corumque Usus, &c.; and in the following year he published a supplement to this work. In the preface to this last he says, that several of the professors of mathematics in Upper Germany, and more especially those who were advanced in years, and grown averse to new methods of reasoning which carried them out of their old principles and habits, doubted whether Napier's demonstration of the property of logarithms was perfectly true, and whether the application of them to trigonometrical calculations might not be unsafe, and lead the calculator who should trust in them to erroneous results; and in either case, whether the doctrine were true or not, they scarcely considered Napier's demonstration of it as legitimate and satisfactory. This opinion induced Kepler to compose the above-mentioned work, in which the whole doctrine is treated in a manner strictly geometrical, and free from the considerations of motion to which those elderly Germans had objected.

On the publication of Napier's Logarithms, Mr Henry Briggs, some time professor of geometry in Gresham College, London, and afterwards Savilian professor of geometry at Oxford, applied himself with great earnestness to the study and improvement of them. From the particular view which Napier took of the subject, and the manner in which he conceived logarithms to be generated, it happened that in his system the logarithms of a series of numbers which increased in a decuple ratio (as 1, 10, 100, 1000, &c.) formed a decreasing arithmetical series, whose common difference was $2\cdot305853$. But it occurred to Briggs that it would be better and more conformable to the received decimal notation, to adopt a system in which the logarithms of the terms of such a geometrical series should differ from each other by unity or 1. This idea Briggs communicated to the public in his lectures, and also to Napier himself. He even went twice to Edinburgh to converse with him on the subject; and, on his first visit, Napier said that he had also formerly thought of the same improvement, but that he chose to publish the logarithms he had previously calculated, till such time as his health and convenience would allow him to make others more commodious. And whereas, in the change which Briggs proposed, it was intended to make the logarithms of the sines to increase from 0 (the logarithm of radius) to infinity, whilst the sines themselves should decrease, it was suggested to him by Napier, that it would be better to make them increase, so that 0, instead of being the logarithm of radius, should be the logarithm of 1, and that 100,000, &c. should be the logarithm of radius. This Briggs admitted would be an improvement; and having changed the numbers he had already calculated so as to make them suit Napier's modification of his plan, he returned with them next year to Edinburgh, and submitted them to his perusal.

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

On Briggs's return from Edinburgh, in 1617, he printed the first thousand logarithms to eight places of figures, besides the index, with the title of *Logarithmorum Chilias prima*; but these seem not to have been published till after the death of Napier, for in his preface he expresses a hope that the circumstances which led to a change in the system would be explained in Napier's posthumous work, about to appear. But although Napier had intimated in a note he had given in Wright's translation of the *Canon Mirificus*, as well as in his *Rabologia*, printed in 1617, that he intended to alter the scale, yet he does not state that Briggs was the first to think of this improvement, or to publish it. And as nothing was said on this point in Napier's posthumous work published in 1619 by his son, Briggs took occasion, in his *Arithmetica Logarithmica*, to assert his claims to the improvement which he had carried into execution. But he has by no means proved that he himself, and not Napier, was the first who had thought of such improvements.

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

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

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

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

Briggs himself lived also to complete a table of logarithmic sines and tangents, to the 100th part of every degree to fourteen places of figures, besides the index, together with a table of natural sines to the same parts to fifteen places, and the tangents and secants of the same to ten places, with the construction of the whole. But death prevented him from completing the application and uses of them. However, when dying, he committed this to his friend Henry Gellibrand, who accordingly added a preface, and the application of the logarithms to plane and spherical trigonometry. The work was called *Trigonometria Britannica*, and was printed at Gouda in 1633, under the care of Adrian Vlacq, who in the same year printed his own *Trigonometria Artificialis, sine Magnus Canon Triangulorum Logarithmicus ad Decadas Secundorum Speculorum Constructus*. This contains the logarithmic sines and tangents to 10 places of figures, with their differences for every ten seconds in the quadrant. It also contains Briggs's table of the first 20,000 logarithms to ten places, besides the index, with their differences; and to the whole is prefixed a description of the tables and their applications. Gellibrand also published, in 1693, *An Institution Trigonometrical*, containing the logarithms of the first 10,000 numbers, with the natural sines, tangents, and secants, and the logarithmic sines and tangents, for degrees and minutes; all to seven decimal places.

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

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

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

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

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

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

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

There are various smaller sets of tables; but probably the most accurate of all are those which Professor Babbage has produced with his very ingenious calculating machine, which has enabled him to detect a variety of errors in former tables. But, what is rather amusing, on examining a set of tables printed in the Chinese character, and which, like every Chinese invention, were older than the deluge, Mr Babbage found they contained precisely the same errors as those of Vlacq did; thus proving, as had long been suspected, from what source those original inventors had derived their logarithms.

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

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

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

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

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

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

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

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

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

II.—NATURE OF LOGARITHMS, AND THEIR CONSTRUCTION.

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

The first step to be taken in constructing a system of logarithms is to assume the logarithm of some determinate number, besides that of unity or 1, which must necessarily be 0. From the particular view which Napier first took of the subject, he was led to assume unity for the logarithm of the number 2·718282, by which it happened that the logarithm of 10 was 2·302585; and this assumption being made, the form of the system became determinate, and the logarithm of every number fixed to one particular value.

It was, however, soon observed, that it would be better to assume unity for the logarithm of 10, instead of making it the logarithm of 2·718282, as in Napier's first system; and hence the logarithms of the terms of the geometrical progression

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

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

\[0, 1, 2, 3, 4, \ldots\] That is, the logarithm of 1 being 0, and that of 10 being 1, the logarithm of 100 is 2, that of 1000 is 3, and so on.

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

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

First step of the process.—The number 5 is between 1 and 10, the logarithms of which we already know to be 0 and 1; Let a geometrical mean be found between the two former, and an arithmetical mean between the two latter. The geometrical mean will be the square root of the product of the numbers 1 and 10, which is \( \sqrt{3} = 162277 \); and the arithmetical mean will be half the sum of the logarithms 0 and 1, which is 0.5; therefore the logarithm of \( \sqrt{3} = 162277 \) is 0.5. But as the mean thus found is not sufficiently near the proposed number, we must proceed with the operation as follows:

Second step.—The number 5, whose logarithm is sought, is between \( \sqrt{3} = 162277 \), 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 \( \sqrt{(3 \times 162277 \times 10)} = 5623413 \), and the other is \( \frac{1 + 0.5}{2} = 0.75 \), the logarithm of 5 = 623413.

Third step.—We have now obtained two numbers, namely, \( \sqrt{3} = 162277 \) and 5623413, one on each side of 5, together with their logarithms 0.5 and 0.75; we therefore, proceeding as before, find the geometrical mean, or \( \sqrt{(3 \times 162277 \times 5623413)} \), to be 4216964, and the arithmetical mean \( \frac{0.5 + 0.75}{2} = 0.625 \), the logarithm of 4216964.

Fourth step.—We proceed in the same manner with the numbers 4216964, and 5623413 (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. 4869674, and the corresponding arithmetical mean or logarithm 0.6875.

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

| Numbers | Logarithms | |---------|------------| | A | 1.000000 | 0.0000000 | | B | 10.000000 | 1.0000000 | | C | \( \sqrt{AB} = 3.162277 \) | 0.5000000 | | D | \( \sqrt{BC} = 5.623413 \) | 0.7500000 | | E | \( \sqrt{CD} = 4.216964 \) | 0.6250000 |

As the last of these means, viz. Z, agrees with 5, the proposed number, as far at least as the sixth place of decimals, we may safely consider them as very nearly equal, and therefore their logarithms very nearly equal; that is, the logarithm of 5 will be 0.689700 nearly.

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

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

From this last remark it is obvious, that having now found the logarithm of 5, we can immediately find that of 2; for since 2 is the quotient of 10 divided by 5, its logarithm will be the difference of the logarithms of 10 and 5; now the logarithm of 10 is 1, and that of 5 is 0.689700, therefore the logarithm of 2 is 0.3010300.

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

Besides the view we have hitherto taken of the theory of logarithms, there are others under which it has been presented by different authors. Some of these we proceed to explain, beginning with that in which they are defined to be the measures of ratios; but to see the propriety of this definition, it must be understood what is meant by the measure of a ratio. According to the usual definition of a compound ratio, if there be any number of magnitudes A, B, C, D, in continued proportion, the ratio of the first, A, to the third, C, is considered as made up of two ratios, each equal the ratio of the first, A, to the second, B. And in like manner the ratio of the first, A, to the fourth, D, is considered as made of three ratios, each equal the same ratio of the first to the second, and so on. Thus, to take a particular example in numbers, because the ratio of 81 to 3 may be considered as made up of the ratio of 81 to 27, and of 27 to 9, and of 9 to 3, which three ratios are equal among themselves, the ratio of 81 to 3 will be triple that of 9 to 3; and in like manner the ratio of 27 to 3 will be double that of 9 to 3. Also, because the ratios of 1000 to 100, 100 to 10, 10 to 1, are all equal, the ratio of 1000 to 1 will be three times as great as that of 10 to 1; and the ratio of 100 to 1 will be twice as great; and so on.

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

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

Let us now suppose that unity, or 1, is assumed as the common consequent or second term of all ratios whatever; and that the ratio of 10 (or some particular number) to 1 is compounded of a very great number of equal ratios, as, for example, 1,000,000; then, as each of these will be very near the ratio of equality (for it will be the ratio of the first term to the second of a series consisting of one million and one continued proportionals, the first of which is 10 and last 1), it will follow, and is easy to conceive, that the ratios of all other numbers to unity will each be very nearly equal to some multiple of that small ratio. And by supposing the number of small equal ratios of which the ratio of 10 to 1 is composed, to be sufficiently great, the ratios of all other numbers to unity may be as nearly equal to ratios which are multiples of that small ratio, as we please. Let us still suppose, however, for the sake of illustration, that the number of small ratios contained in that of 10 to 1 is 1,000,000; then, as it may be proved that the ratio of 2 to 1 will be very nearly the same as a ratio composed of 301,030 of these, and that the ratio of 3 to 1 will be nearly equal to a ratio composed of 477,121 of them; and that the ratio of 4 to 1 will be nearly equal to a ratio composed of 602,060 of them, and so on; these numbers, viz. 1,000,000, 301,030, 477,121, and 602,060, or any other numbers proportional to them, will be the measures of the ratios of 10 to 1, 2 to 1, 3 to 1, and 4 to 1, respectively; and the same quantities will also be what have been called the logarithms of the ratios; for the word logarithm, if regard be had to its etymology, is ἀριθμός ἀριθμοῦ, or the numbers of small and equal ratios (or ratios unceles, as they have been called) contained in the several ratios of quantities one to another.

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

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

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

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

The common scale of notation in arithmetic is so contrived as to express all numbers whatever by the powers of 10, which is the root of the scale, and the nine digits serving as co-efficients to these powers. Thus, if R denote 10, the root of the scale, so that \(R^2\) will denote 100, and \(R^3\) 1000, and so on, the number 471,509 is otherwise expressed by \(4R^5 + 7R^4 + 1R^3 + 5R^2 + 0R^1 + 9R^0\), which is Let \( n \) express any number whatever; then raising both sides of the equation \( a = r^A \) to the \( n \)th power, we have \( a^n = (r^A)^n = r^{An} \); but here \( A \) is manifestly the logarithm of \( a^n \); therefore, the logarithm of \( a^n \), any power of a number, is the product of the logarithm of the number by \( n \), the index of the power. This must evidently be true, whether \( n \) be a whole number or a fraction, positive or negative.

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

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

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

If we suppose \( r \) and \( a \) to be each a positive number greater than unity, then \( A \) will be a positive number; for if it be a negative we would have \( a = r^{-A} = \frac{1}{r^A} \), a proper fraction, and at same time a number greater by hypothesis than unity, which is absurd. If, on the contrary, we suppose \( a \) a proper fraction, then \( A \) must necessarily be negative, otherwise \( r^A \) would be greater than unity, and \( a = r^A \) also greater than unity, while by hypothesis it is a fraction less than unity, which is absurd. Therefore, in every system, the base of which exceeds unity, the logarithm of a whole or mixed number is always positive, but the logarithm of a proper fraction is always negative.

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

Let \( r \) and \( r' \) denote bases of two different systems; and let \( A \) be the logarithm of a number, \( a \), taken according to the first of these, and \( A' \) its logarithm according to the second. Then, because \( a = r^A \), and \( a = r'^{A'} \), it follows that

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

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

\[ \begin{align*} y^R &= r, \\ r'^R &= r'; \end{align*} \]

we have \( r^R R' = r' R, r' R R' = r^R \);

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

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

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.

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

\[ \begin{align*} r^A &= a, \\ r^B &= b; \end{align*} \]

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

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

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

\[ y = r^x. \]

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

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

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

\[ \begin{align*} 1 &= 1, \\ xa &= + ax, \\ \frac{x(x-1)}{1 \cdot 2} a^2 &= - \frac{a^2}{2} x + \frac{a^2}{2} x^2, \\ \frac{x(x-1)(x-2)}{1 \cdot 2 \cdot 3} a^3 &= + \frac{a^3}{3} x - \frac{a^3}{2} x^2 + \frac{a^3}{6} x^3, \\ \frac{x(x-1)(x-2)(x-3)}{1 \cdot 2 \cdot 3 \cdot 4} a^4 &= - \frac{a^4}{4} x + \frac{11a^4}{24} x^2 - \frac{a^4}{4} x^3 + \frac{a^4}{24} x^4, \\ &\vdots \end{align*} \]

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

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

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

\[ \begin{align*} A' &= \frac{a^2}{2} - \frac{a^3}{2} + \frac{11a^4}{24} - \cdots, \\ A'' &= \frac{a^3}{6} - \frac{a^4}{4} + \cdots, \\ A''' &= \frac{a^4}{24} - \cdots, \end{align*} \]

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

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

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

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

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

for the same reason

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

therefore the series

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

is equal to the product of the two series

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

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

\[ \begin{align*} 1 + Ax + A'x^2 + A''x^3 + A'''x^4 + \cdots \\ + Az + 2Axz + 3A'xz^2 + 4A''xz^3 + \cdots \\ + A'z^2 + 3A''z^3 + 6A'''z^4 + \cdots \\ + A''z^3 + 4A'''z^4 + \cdots \\ + A'''z^4 + \cdots \end{align*} \]

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

\[ \begin{align*} A' &= \frac{A^2}{1 \cdot 2}, \\ AA' &= 3A'', \\ AA'' &= 4A''', \end{align*} \]

and hence

\[ \begin{align*} A' &= \frac{A^2}{1 \cdot 2}, \\ A'' &= \frac{A^3}{1 \cdot 2 \cdot 3}, \\ A''' &= \frac{A^4}{1 \cdot 2 \cdot 3 \cdot 4}, \end{align*} \]

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

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

and it becomes

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

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

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

is otherwise expressed by

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

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

If in the formula

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

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

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

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

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

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

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

\[ 27182818284590452353602874. \]

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

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

is the logarithm of \( r \) to the base \( e \). But as \( r \) is not 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

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

supposing that the base of the system is the number we have expressed above by \( e \).

We have now found a general formula for the logarithm of any number \( y \), taken according to a particular system, namely, that which has the number \( e \) for its base. But it is easy from hence to find a formula, which shall apply to any system whatever. For it has been shown that the 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,

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

hence we find

\[ \log_y : \log_r = \frac{\log_e}{\log_r} \times \log_y : \log_e. \]

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

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

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

If in the above formula we suppose \( r = e \), the multiplier \( \frac{\log_e}{\log_r} \) will be unity, and the formula will become simply

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

as we have already remarked. Now this is the system which was adopted by Lord Napier; and although the logarithms which were computed according to this system, or upon the supposition that the radical number is 27182818, &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 \( \log_r \) to base \( r \); and the denominator will be unity, so that the modulus of any system whose base is \( r \), is the reciprocal of the Napierian logarithm of that base; or it is the logarithm of the number \( e \) (the base of the Napierian system) to the base \( r \).

In the Napierian system, the modulus is unity, and hence the logarithms of this system are more easily computed than those of any other. It was, however, soon found that a system whose base should be the same as the root of the scale of the arithmetical notation, viz. the number 10, would be the most convenient of any in practice; and accordingly such a system was actually constructed by Mr Briggs. This is the only one now in common use, and is called Briggs's system, also the common system of logarithms. The modulus of this system therefore is the reciprocal of the Napierian logarithm of 10, viz. 4342948, which is the common logarithm of \( e = 27182818 \), &c. the base of the Napierian system. We shall in future denote this modulus by \( M \); so that the formula expressing the common logarithm of any number \( y \) will be

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

If the number \( y \), whose logarithm is required, be very near Construc- to unity, so that \(1 - y\) is a small quantity, then the loga- rithm may be found from this formula with great ease, because the series will converge very rapidly. If, how- ever, \(1 - y\) be greater than unity, the series, instead of converging, will diverge, so as to be, in its present form, of no use.

It may, however, be transformed into another, which shall converge in every case, by substituting in it \(n\sqrt{y}\) instead of \(y\), and observing that \(\log_y = \frac{\log_y}{n}\); it thus becomes

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

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

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

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

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 if \(y\) be greater than unity, \(n\sqrt{y}\) will also be greater than unity, and vice versa; but the differences will be so much the smaller as \(n\) the exponent of the root is greater; therefore \(n\sqrt{y} - 1\) will be positive in the first case, and negative in the second.

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

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

also

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

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 then the expression \(1 - \frac{1}{n\sqrt{y}}\) will also be less than unity, seeing it is equal \(\frac{n\sqrt{y} - 1}{n\sqrt{y}}\). Therefore, in the first series, the second and third terms (taken together as one term) con- stitute a negative quantity; and as the same is also true of the fourth and fifth, and so on, the amount of all the terms after the first is a negative quantity, or one which is to be subtracted from the first, to obtain the value of \(\log_y\). Hence

\[ \log_y < nM(\sqrt{y} - 1). \]

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

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

Thus we have two limits to the value of the logarithm of \(y\), which, by taking the number \(n\) sufficiently great, may come as near to each other as we please.

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

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

It is evident that the difference between the two limits of \(\log_y\) is

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

therefore, if we take either the one or the other of the two preceding expressions for \(\log_y\), the error in excess or de- fect is necessarily less than this quantity.

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

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

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 extrac- tions of the square root only. It was in this way that Briggs calculated the first logarithms; and he remarked, that if in performing the successive extractions of the square root, he at last obtained twice as many decimal places as there were ciphers after the decimal point, the integer before it being unity, then the decimal part of this root was exactly the half of that which went before; so that the decimal parts of the two roots were to each other in the same proportion as their logarithms: now this is an evident consequence of the preceding formula.

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

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

when \(n\) is some very great number. Let us suppose \(n = 2^{60} = 8^{30}\); then, dividing unity by 8, and this result again by 8, and so on, we shall, after 20 divisions, have

\[ \frac{1}{n}, \quad \frac{1}{8^{30}}, \text{ equal to } \]

0 000000 000000 000000 00086736173798840354.

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

1 000000 000000 000000 0016971742081255052703251.

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

86736173798840354 = 0.4342944819.

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

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

\(n\) 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 \(n\sqrt{3}\) equal to

1 000000 000000 000000 0095289426407458932. Therefore, taking the value of \( \sqrt[3]{10} \) as found in last example, we have

\[ \log_3 = \frac{\sqrt[3]{3} - 1}{\sqrt[3]{10} - 1} = \frac{95289425407458932}{199717420812550527} = 477121254719662. \]

This method of computing logarithms is evidently attended with great labour, on account of the number of extractions of roots which it requires to obtain a result true to a moderate number of places of figures. But the two series which we have given serve to simplify and complete it. For, whatever be the number \( y \), it is only necessary to proceed with the extractions of the square root, till we have obtained for \( \sqrt[3]{y} \) a value which is unity followed by a decimal fraction; and then \( \sqrt[3]{y} - 1 \) being a fraction, its powers will also be fractions so much the smaller as their exponents are greater; thus a certain number of terms of the series will express the logarithm to as many decimal places as may be required.

There are yet other artifices by which the series

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

may be transformed into others which shall always converge, and in particular the following. Let \( 1 + u \) be substituted in the series for \( y \); then it becomes

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

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

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

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

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

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

\[ \log_y = 2M\left(\frac{z-1}{z+1} + \frac{1}{3}\left(\frac{z-1}{z+1}\right)^3 + \frac{1}{5}\left(\frac{z-1}{z+1}\right)^5 + \ldots\right); \]

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

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

\[ \frac{2}{3} + \frac{1}{3} + \frac{1}{7} + \frac{1}{9} + \ldots \]

where \( A \) is put for \( \frac{2}{3} \), \( B \) for \( \frac{1}{3} \), \( C \) for \( \frac{1}{7} \), \( D \) for \( \frac{1}{9} \), \( E \) for \( \frac{2}{3} \), \( F \) for \( \frac{1}{3} \), \( G \) for \( \frac{1}{7} \), \( H \) for \( \frac{1}{9} \), \( I \) for \( \frac{2}{3} \), \( J \) for \( \frac{1}{3} \), \( K \) for \( \frac{1}{7} \), \( L \) for \( \frac{1}{9} \), \( M \) for \( \frac{2}{3} \), \( N \) for \( \frac{1}{3} \), \( O \) for \( \frac{1}{7} \), \( P \) for \( \frac{1}{9} \).

The calculation will be as follows:

\[ A = 666666666666 \\ B = 0.024691358025 \\ C = 0.01646090535 \\ D = 0.00130642106 \\ E = 0.00011290056 \\ F = 0.00001026369 \\ G = 0.00000098496 \\ H = 0.00000009292 \\ I = 0.00000000911 \\ J = 0.00000000091 \\ K = 0.00000000001 \\ L = 0.00000000000 \\ M = 0.00000000000 \]

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 0.693147180556.

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

In the same manner as we have found the logarithm of 2, we may find those of 3, 5, &c. In computing the logarithm of 3, the series would converge by the powers of the fraction \( \frac{3-1}{3+1} = \frac{1}{2} \), 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 \( \frac{1+u}{1-u} = 1 + \frac{z}{n} \) and consequently \( u = \frac{z}{2n+z} \), these values being substituted in the formula, \( \log_y \frac{1+u}{1-u} = 2M\left(u + \frac{u^3}{3} + \frac{u^5}{5} + \ldots\right) \)

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

\[ 2M\left(\frac{z}{2n+z} + \frac{1}{3}\left(\frac{z}{2n+z}\right)^3 + \frac{1}{5}\left(\frac{z}{2n+z}\right)^5 + \ldots\right) \]

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

therefore, \( \log_y(n+z) = \log_y n + 2M\left(\frac{z}{2n+z} + \frac{1}{3}\left(\frac{z}{2n+z}\right)^3 + \frac{1}{5}\left(\frac{z}{2n+z}\right)^5 + \ldots\right) \)

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

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

Therefore the logarithm of 3 is

\[ \log_2 + 2\left(\frac{1}{5} + \frac{1}{3} + \frac{1}{7} + \frac{1}{9} + \ldots\right) \]

\[ = \log_2 + A + \frac{1}{3}B + \frac{1}{3}C + \frac{1}{7}D + \frac{1}{9}E + \ldots \] The calculation may stand thus:

\[ \begin{align*} A &= \frac{1}{2} = 0.49999999999 \\ B &= \frac{1}{3} A = 0.16666666667 \\ C &= \frac{1}{5} B = 0.02000000000 \\ D &= \frac{1}{7} C = 0.002857143 \\ E &= \frac{1}{9} D = 0.000285714 \\ F &= \frac{1}{11} E = 0.000025641 \\ G &= \frac{1}{13} F = 0.000001930 \\ H &= \frac{1}{15} G = 0.000000129 \\ I &= \frac{1}{17} H = 0.000000075 \\ J &= \frac{1}{19} I = 0.000000041 \\ K &= \frac{1}{21} J = 0.000000024 \\ L &= \frac{1}{23} K = 0.000000014 \\ M &= \frac{1}{25} L = 0.000000007 \\ N &= \frac{1}{27} M = 0.000000004 \\ O &= \frac{1}{29} N = 0.000000002 \\ P &= \frac{1}{31} O = 0.000000001 \\ Q &= \frac{1}{33} P = 0.000000000 \\ R &= \frac{1}{35} Q = 0.000000000 \\ S &= \frac{1}{37} R = 0.000000000 \\ T &= \frac{1}{39} S = 0.000000000 \\ U &= \frac{1}{41} T = 0.000000000 \\ V &= \frac{1}{43} U = 0.000000000 \\ W &= \frac{1}{45} V = 0.000000000 \\ X &= \frac{1}{47} W = 0.000000000 \\ Y &= \frac{1}{49} X = 0.000000000 \\ Z &= \frac{1}{51} Y = 0.000000000 \\ \end{align*} \]

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

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

Because \(2 \times \frac{5}{7} = \frac{50}{49}\), therefore log. 2 + 2 log. 5 - 2 log. 7 = log. \(\frac{50}{49}\) and consequently

\[ \log. 7 = \frac{1}{2} \log. 2 + \log. 5 - \frac{1}{2} \log. \frac{50}{49}. \]

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

\[ \log. z = 2M \left( \frac{z-1}{z+1} + \frac{1}{3} \left( \frac{z-1}{z+1} \right)^3 + \frac{1}{5} \left( \frac{z-1}{z+1} \right)^5 + \text{etc.} \right) \]

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

\[ \log. \frac{50}{49} = 2 \left( \frac{1}{99} + \frac{1}{399} + \frac{1}{599} + \text{etc.} \right) \]

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

\[ \begin{align*} A &= 0.02020202020 \\ B &= \frac{1}{9 \cdot 11} = 0.00000206122 \\ C &= \frac{1}{9 \cdot 11} = 0.00000000021 \\ D &= \frac{1}{9 \cdot 11} = 0.00000000004 \\ E &= \frac{1}{9 \cdot 11} = 0.00000000000 \\ F &= \frac{1}{9 \cdot 11} = 0.00000000000 \\ G &= \frac{1}{9 \cdot 11} = 0.00000000000 \\ H &= \frac{1}{9 \cdot 11} = 0.00000000000 \\ I &= \frac{1}{9 \cdot 11} = 0.00000000000 \\ J &= \frac{1}{9 \cdot 11} = 0.00000000000 \\ K &= \frac{1}{9 \cdot 11} = 0.00000000000 \\ L &= \frac{1}{9 \cdot 11} = 0.00000000000 \\ M &= \frac{1}{9 \cdot 11} = 0.00000000000 \\ N &= \frac{1}{9 \cdot 11} = 0.00000000000 \\ O &= \frac{1}{9 \cdot 11} = 0.00000000000 \\ P &= \frac{1}{9 \cdot 11} = 0.00000000000 \\ Q &= \frac{1}{9 \cdot 11} = 0.00000000000 \\ R &= \frac{1}{9 \cdot 11} = 0.00000000000 \\ S &= \frac{1}{9 \cdot 11} = 0.00000000000 \\ T &= \frac{1}{9 \cdot 11} = 0.00000000000 \\ U &= \frac{1}{9 \cdot 11} = 0.00000000000 \\ V &= \frac{1}{9 \cdot 11} = 0.00000000000 \\ W &= \frac{1}{9 \cdot 11} = 0.00000000000 \\ X &= \frac{1}{9 \cdot 11} = 0.00000000000 \\ Y &= \frac{1}{9 \cdot 11} = 0.00000000000 \\ Z &= \frac{1}{9 \cdot 11} = 0.00000000000 \\ \end{align*} \]

This logarithm is true to 10 decimal places, the accurate value to 12 figures being 1.0986122886868, which would have been our result had the correct value of log. 2, viz. 0.693147180551 been employed.

The Napierian logarithm of 4 is immediately had from that of 2 by considering that as 4 = 2², therefore log. 4 = log. 2 + log. 2.

\[ \log. 2 = 0.693147180551 \]

\[ \log. 4 = 1.386294361102 \]

This is also true to 10 places besides the integer.

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

\[ \log. 4 + \frac{1}{2} \left( \frac{1}{9} + \frac{1}{39} + \frac{1}{59} + \frac{1}{79} + \text{etc.} \right) \]

where \(A = \frac{1}{9}\), \(B = \frac{1}{39}\), \(C = \frac{1}{59}\), \(D = \frac{1}{79}\), etc.

The calculation.

\[ \begin{align*} A &= 0.29999999999 \\ B &= \frac{1}{9} A = 0.02743484925 \\ C &= \frac{1}{39} B = 0.000033870176 \\ D &= \frac{1}{59} C = 0.000000418150 \\ E &= \frac{1}{79} D = 0.000000005162 \\ F &= \frac{1}{99} E = 0.000000000064 \\ G &= \frac{1}{119} F = 0.00000000000 \\ H &= \frac{1}{139} G = 0.00000000000 \\ I &= \frac{1}{159} H = 0.00000000000 \\ J &= \frac{1}{179} I = 0.00000000000 \\ K &= \frac{1}{199} J = 0.00000000000 \\ L &= \frac{1}{219} K = 0.00000000000 \\ M &= \frac{1}{239} L = 0.00000000000 \\ N &= \frac{1}{259} M = 0.00000000000 \\ O &= \frac{1}{279} N = 0.00000000000 \\ P &= \frac{1}{299} O = 0.00000000000 \\ Q &= \frac{1}{319} P = 0.00000000000 \\ R &= \frac{1}{339} Q = 0.00000000000 \\ S &= \frac{1}{359} R = 0.00000000000 \\ T &= \frac{1}{379} S = 0.00000000000 \\ U &= \frac{1}{399} T = 0.00000000000 \\ V &= \frac{1}{419} U = 0.00000000000 \\ W &= \frac{1}{439} V = 0.00000000000 \\ X &= \frac{1}{459} W = 0.00000000000 \\ Y &= \frac{1}{479} X = 0.00000000000 \\ Z &= \frac{1}{499} Y = 0.00000000000 \\ \end{align*} \]

This result is also correct to the first ten places.

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

\[ \frac{2}{3} = 0.666666667 \]

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

\[ M = 4.34294481903251827651128918916 \] \[ \frac{1}{M} = 2.302585092994045684017991454684 \]

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

Because

\[ \log z = 2M \left\{ \frac{z-1}{z+1} + \frac{1}{3} \left( \frac{z-1}{z+1} \right)^3 + \frac{1}{5} \left( \frac{z-1}{z+1} \right)^5 + \cdots \right\} \]

If we now suppose

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

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

\[ \log_z = 2M \left\{ \frac{1}{2n^2 - 1} + \frac{1}{3} \left( \frac{1}{2n^2 - 1} \right)^3 + \frac{1}{5} \left( \frac{1}{2n^2 - 1} \right)^5 + \cdots \right\} \]

But \( \log_z = \log_n - \log(n-1) - \log(n+1) \), therefore, putting N for the series

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

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

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

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

where \( N = \frac{2M}{199} + \frac{2M}{3 \cdot 199^2} + \cdots \)

\( M \) being \( 4.3429448190 \).

Calculation of N.

\[ A = \frac{2M}{199} = 0.00436476866 \] \[ B = \frac{A}{3 \cdot 199^2} = 0.00000003674 \] \[ N = 0.00436480540 \] \[ \log_9 = 0.95424250943 \] \[ \log_{10} = 1 \] \[ \log_{11} = 1.04139268517 \]

Here the series expressed by N converges very fast, so that two of its terms are sufficient to give the logarithm of Lo. true to 10 places of decimals. But the logarithm of 11 logarithms 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 \( 98 = 2 \times 7 \), \( 99 = 9 \times 11 \), 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 \( \log_{99} \), \( \log_2 + \log_7 \) for \( \log_{98} \), and \( \log_{10} + \log_{10} \) for \( \log_{100} \),

\[ 2 \log_9 + 2 \log_{11} - \log_2 - 2 \log_7 - 2 \log_{10} = N, \]

and hence by transposition, &c.

\[ \log_{11} = \frac{1}{2} N + \frac{1}{2} \log_2 + \log_7 - \log_9 + \log_{10}; \]

and in this equation

\[ N = \frac{2M}{19601} + \frac{2M}{3 \cdot 19601^2} + \cdots \]

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

\[ \log_z = 2M \left\{ \frac{z-1}{z+1} + \frac{1}{3} \left( \frac{z-1}{z+1} \right)^3 + \frac{1}{5} \left( \frac{z-1}{z+1} \right)^5 + \cdots \right\} \]

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

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

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

\[ \log_z = 2M \left\{ \frac{2}{n^2 - 3n} + \frac{1}{3} \left( \frac{2}{n^2 - 3n} \right)^3 + \frac{1}{5} \left( \frac{2}{n^2 - 3n} \right)^5 + \cdots \right\} \]

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,

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

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,

\[ \log_7 + 2 \log_2 - 3 \log_3 = \frac{2M}{55} + \frac{2M}{3 \cdot 55^2} + \cdots \] \[ -2 \log_7 + \log_2 + 2 \log_5 = \frac{2M}{99} + \frac{2M}{3 \cdot 99^2} + \cdots \] \[ 4 \log_3 - 4 \log_2 - \log_5 = \frac{2M}{161} + \frac{2M}{3 \cdot 161^2} + \cdots \] \[ \log_5 - 5 \log_3 + 2 \log_7 = \frac{2M}{244} + \frac{2M}{3 \cdot 244^2} + \cdots \]

Let log_2, log_3, log_5, and log_7, be now considered as Construct four unknown quantities, and by resolving those equations in the usual manner, the logarithms may be determined.

Resuming once more the formula

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

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

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

and the formula will be transformed to

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

James Gregory, Vera Circuli et Hyperbolae Quadratura, 1667.

James Gregory, Exercitationes Geometricæ, 1668.

James Gregory, in Commercium Epistolicum, 1712.

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

Brouncker and Wallis, Phil. Trans. 1668.

Barrow, Lectiones Geometricæ, 1674.

Halley, Phil. Trans. 1695, 1696.

Jones, Synopsis Mathematicæ, 1706.

Craig, Phil. Trans. 1710.

Sir Isaac Newton, Commercium Epistolicum.

Sir Isaac Newton, Method of Fluxions, 1736.

Cotes, Phil. Trans. 1714.

Cotes, Harmonia Mensurarum, 1722.

Long, Phil. Trans. 1714.

Brooke Taylor, Phil. Trans. No. 352.

Dodson, Phil. Trans. 1753.

Masceres, Trigonometry, 1760.

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

Waring, Phil. Trans. 1779.

Hellins, Phil. Trans. 1780-1796.

Lagrange, Théorie des Fonctions.

Laplace, Journal de l'Ecole Polytechnique, 1809.

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

Lavermède, Annales des Mathématiques, 1811.

Vincent, Annales des Mathématiques, 1824, 1825.

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

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

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

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

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

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

To find the limits of approximation, let three terms of the fractional series be taken, and $\sqrt{\left(\frac{a^2 + 1}{a^2}\right)} = \frac{8a^4 + 8a^3 + 1}{8a^4 + 4a^2}$, or $\sqrt{\left(\frac{b+2}{b}\right)} = \frac{2b^2 + 4b + 1}{2b^2 + 2b}$

$$= \frac{2b + 3}{2b + 1} \left(1 + \frac{1}{(2b + 3)(2b + 1)b}\right).$$ Whence $\frac{1}{2} \log \left(\frac{b+2}{b}\right) = \frac{25 + 3}{2b + 1} + 1 \left(1 + \frac{1}{(2b + 3)(2b + 1)b}\right)$;

and, therefore, since this last quantity exceeds unit only by a very minute difference, $\frac{1}{2} \log \frac{b+2}{2} = \log \frac{2b + 3}{2b + 1}$

$$+ \frac{M}{(2b + 3)(2b + 1)b},$$ where $M$ denotes the modulus of the system. If the number $2b + 3$ or $2b + 1$ be ex- pressed by N; this small correction will amount but to \( \frac{M}{N^3} \) or \( \frac{2M}{N^3} \). Consequently \( \log_2 \frac{b+3}{2b+1} = \frac{1}{2} \log_b \frac{b+2}{b} \).

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

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

\[ \begin{array}{ccc} 35150 & -2 & 23151 \\ 30706 & -3 & 22143 \\ 27899 & -4 & 21290 \\ 25898 & -5 & 20557 \\ 24372 & -6 & 19913 \\ \end{array} \]

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

| Numbers | Logarithms | Alternate Differences | Their Halves | |---------|------------|-----------------------|--------------| | 150 | 2.1760913 | | | | 151 | 2.1789769 | | | | 152 | 2.1818436 | 57523 | 28761 | | 153 | 2.1846914 | 57145 | 28572 | | 154 | 2.1875207 | 56771 | 28386 | | 155 | 2.1903317 | 56403 | 28201 |

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

In this manner the operation may be continued; but, to prevent any accumulation of errors, the logarithms of the composite numbers should serve as standards, being formed by the addition of the logarithms of their several factors. The logarithms of the intermediate even numbers 302, 304, 306, 308, and 310, are easily determined by adding .3010500 to the logarithms of 151, 152, 153, 154, and 155.

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

| Numbers | Logarithms | Alternate Differences | Their Halves | |---------|------------|-----------------------|--------------| | 2000 | 3.3010299957 | | | | 2001 | 3.3014270886 | | | | 2002 | 3.3014640731 | 4340774 | 2170387 | | 2003 | 3.3016509493 | 4338607 | 2169303 | | 2004 | 3.3018977172 | 4336441 | 2168220 | | 2005 | 3.3021143770 | 4334277 | 2167139 |

Whence are derived,

| Numbers | Logarithms | |---------|------------| | 4001 | 3.8021885514 | | | 2170387 | | 4003 | 3.6023855901 | | | 2169303 | | 4005 | 3.6026025204 | | | 2168220 | | 4007 | 3.6028193424 | | | 2167139 | | 4009 | 3.6030360563 |

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

| Numbers | Logarithms | Alternate Differences | Their Halves | |---------|------------|-----------------------|--------------| | 9995 | 3.99978279845413 | | | | 9996 | 3.99982621745441 | | | | 9997 | 3.999869692108927 | | | | 9998 | 3.99991313241658 | | | | 9999 | 3.99995656838020 | | | | 10000 | 4.00000000000000 | | | | 19991 | 4.30083451916161 | 4344682696 | | | 19993 | 4.30087796598857 | 4344248097 | | | 19995 | 4.30092140846954 | 4343813585 | | | 19997 | 4.30096484660539 | 4343379161 | | | 19999 | 4.30100828039700 | | |

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

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

\[ D - B = \frac{E - A}{2}, \text{ whence } D = B + \frac{E - A}{2}. \]

Applying this to the logarithms of eight places of figures, let A, B, C, D, E, &c. represent the logarithms of 500, 501, 502, 503, 504, &c. then log. 503 = log. 501 + log. 504 — log. 500

\[ = \log_2 501 + \frac{\log_2 252 - \log_2 250}{2} = 2.69983773 + \frac{1}{2} \]

(2.40140054 — 2.39794001) = 2.79156799; the halves of the differences of the alternate logarithms of 250, 251, 252, 253, &c. being thus taken, as before, to compose by their additions the logarithms of the odd numbers 503, 505, 507, &c. But since $E - A = 2D - 2B$, it follows that $E = A + 2(D - B)$. Therefore, in any series, the fifth term will be found nearly, by adding to the first term twice the difference between the second and fourth terms. In this way the tables of natural sines, tangents, and secants, could easily be framed. Thus, the logarithmic sines of the successive arcs, $50^\circ$, $50^\circ 1'$, $50^\circ 2'$, and $50^\circ 3'$ being given from Vlacq's Tables, to find the logarithmic sine of $50^\circ 4'$.

| Arcs | Logarithmic Sines | |------|-------------------| | $50^\circ$ | 9-8842599665 | | $1'$ | 9-8843599996 | | $2'$ | 9-8844658502 | | $3'$ | 9-8845716981 | | $4'$ | 9-8846774835 |

Here, passing over the middle term, the difference between the logarithmic sines of $50^\circ 1'$ and of $50^\circ 3'$ is doubled, and added to that of $50^\circ$, to form the logarithmic sine of $50^\circ 4'$.

But a nearer approximation may be obtained, by supposing the second differences of any series to form the arithmetical progression. The sum of the extreme terms $C - 2B + A$ and $F - 2E + D$ would, therefore, be equal to the sum of the mean terms $D - 2C + B$ and $E - 2D + C$; whence $F - A = 3B + 3E - 2C - 2D$, or $F = A + 3(B + E) - 2(C + D)$. Thus, the natural sines of the successive arcs $30^\circ$, $31^\circ$, $32^\circ$, $34^\circ$, and $35^\circ$ degrees may be easily computed to seven places of figures.

| Arcs | Sines | |------|-------| | $30^\circ$ | .5000000 | | $31^\circ$ | .5150381 | | $32^\circ$ | .5299193 | | $33^\circ$ | .5446390 | | $34^\circ$ | .5592929 | | $35^\circ$ | .5735764 |

The sines of $31^\circ$ and of $34^\circ$ are here added together, and the sum tripled, and from this amount is taken twice the sum of the sines of $31^\circ$ and of $33^\circ$; the sine of $30^\circ$ being subtracted from that remainder, leaves finally the sine of $35^\circ$.

Employing the same number of terms of the series, a still closer approximation may be discovered, by considering the third differences only as uniformly progressive. Wherefore the extreme differences $D - 3C + 3B - A$ and $F - 3E + 3D - C$ will be together equal to double the middle one, $E - 3D + 3C - B$, and consequently $F - A = 5E - 10D + 10C - 5B$, or $F = A + 5(E - B) - 10(D - C)$.

Hence the logarithms even of low numbers may be computed exact to eight decimal places. Thus, the logarithms of $150$, $151$, $152$, $153$, and $154$, being given, that of $155$ is found by this process:

| Numbers | Logarithms | |---------|------------| | 150 | 2-17609126 | | 151 | 2-17897695 | | 152 | 2-18189359 | | 153 | 2-18469143 | | 154 | 2-18752072 | | 155 | 2-19033170 |

The difference between the logarithms of $151$ and $154$ is here multiplied by $5$, and the difference of the logarithms of $152$ and $153$ is multiplied by $10$; and the excess of the former product above the latter being added to the logarithm of $150$, gives the logarithm of $155$.

It would obviously be preferable, however, to employ the formula in a modified form for interpolation merely. Hence $D = C + \frac{1}{2}(E - B) + A - F$.

If six terms of the series were given, the seventh could be found to a high degree of accuracy. The sum of the extremes of the progressive third differences being now assumed equal to that of the means, we have $D - 3C + 3B - A + G - 3F + 3E - D = E - 3D + 3C - B + F - 3E + 3D - C$, and by reduction $G = A = 4F - 5E - 5C - 4B$, whence $G = A + 4(F - B) - 5(E - C)$. It seems unnecessary to subjoin any further illustrations; but the very simple methods of interpolation now proposed might be applied with great facility and advantage in various physical researches. In this way much light may be thrown upon the resistance of fluids, and upon the force, the density, and the component heat of steam, at different temperatures.

**LOGIC.**

Logic is the art of properly conducting reason in the knowledge of things, whether for instructing ourselves or others; or it may be defined the science of human thought, inasmuch as it traces the progress of knowledge, from the first and most simple notions, through all the different combinations of these, and the numerous deductions which result from variously comparing them one with another.

The business of logic, therefore, is to evolve the laws of human thought, and the proper manner of conducting the reason, in order to the attainment of truth and knowledge. It lays open those errors and mistakes which we are apt, through inattention, to run into; and teaches us how to distinguish between truth, and what only wears the resemblance thereof. By these means we become acquainted with the nature, limits, and force of the understanding, and see what things lie within its reach; where we may attain certainty and demonstration; and when we must be contented with probability.

This science is generally divided into four parts, namely, Perception, Judgment, Reasoning, and Method.

I.—OF PERCEPTION.

We find ourselves surrounded with a variety of objects, which, acting differently upon our senses, convey different impressions into the mind, and thereby rouse the attention and notice of the understanding. By reflecting, too, on what passes within us, we become sensible of the operations of our own minds, and attend to them as a new set of impressions. But in all this there is only bare consciousness. The mind, without proceeding any further, takes notice of the impressions which are made upon it, and views things in order, as they present themselves one after another. This attention of the understanding to the objects acting upon it, whereby it becomes sensible of the impres- Of Perceptions they make, is called by logicians perception; and the notices themselves, as they exist in the mind, and are there treasured up to become the materials of thinking and knowledge, are distinguished by the names of ideas, notions, thoughts, &c. In the article Metaphysics it will be shown at large, how the mind, being furnished with ideas, contrives to diversify and enlarge its stock; we have here chiefly to consider the means of making known our thoughts to others, that we may not only understand how knowledge is acquired, but also in what manner it may be communicated with the greatest certainty and advantage.

1. Of Words, considered as the Signs of Ideas.

Words the means of thought and various, are nevertheless of themselves unknown recording to others. But God, designing us for society, has provided us with organs fitted to frame articulate sounds, and given us also a capacity of using those sounds as the signs of internal conceptions. Hence spring words and language; for, having once pitched upon any sound to stand as the mark or sign of an idea in the mind, custom by degrees establishes such a connection between them, that the appearance of the idea in the understanding suggests to our remembrance the sound or name by which it is expressed; as, in like manner, the hearing of the sound excites the idea which it represents. And thus it is easy to conceive how a man may record his own thoughts, and bring them again into view in any succeeding period of life; for this connection being once settled, as the same sounds will always serve to excite the same ideas, if he can but contrive to register his words in the order and disposition in which the actual train of his thoughts presents itself to his imagination, it is evident that he will be able to recall these thoughts at pleasure, and that too in the very manner of their first appearance. Accordingly we find, that the inventions of writing and printing, by enabling us to fix and perpetuate such perishable things as sounds, have also furnished us with the means of giving a kind of permanence to the transactions of the mind, insomuch that they may thereby be subjected to our review like any other objects of nature.

2. But besides the capacity of recording our thoughts, external signs also enable us to communicate our thoughts to others, and to receive information of what passes in their minds. For any number of men, having agreed to establish the same sounds as signs of the same ideas, it is apparent that the repetition of these sounds must suggest similar ideas in each, and thus tend to create a perfect correspondence of thoughts. When, for instance, any train of ideas succeed one another in the mind, if the names by which we are wont to express them have been annexed by those with whom we converse to the same order of ideas, nothing is more evident, than that, by repeating those names according to the tenor of our actual conceptions, we shall raise in their minds a similar train of thought. For, by barely attending to what passes within themselves upon hearing the sounds which we repeat, they will also be made aware of the ideas in our understanding. So that we here clearly perceive how a man may communicate his sentiments, knowledge, and discoveries to others, if the language in which he converses be extensive enough to mark all the ideas and operations of his mind. But as this is not always the case, and men are often obliged to invent terms of their own to express new views and conceptions of things, it may be asked, how, in these circumstances, we can become acquainted with the thoughts of another, when he makes use of words with which we have never associated ideas? In order to unveil this mystery, and afford some insight into the foundation, progress, and improvement of language, the following observations are deserving of attention.

3. First, it is evident that no word can be to any man the sign of an idea, unless that idea comes to have a real existence in his mind. For names, being only so far intelligible as they denote known internal conceptions; where they have none such answering to them, they are plainly sounds without signification, and of course they convey no instruction or knowledge. But no sooner are the ideas to which they belong raised in the understanding, than, finding it easy to connect them with the established names, we can join in any agreement of this kind established by others, and thereby enjoy the benefit of their discoveries. The first thing therefore to be considered is, how these ideas may be conveyed into the mind; that being there, we may learn to connect them with their appropriate sounds, and so become capable of understanding others when they make use of these sounds in communicating their thoughts. Now, to comprehend this distinctly, it is necessary to attend to the division of our ideas into simple and complex (see Metaphysics). And, firstly, as to our simple ideas, they can find no admission into the mind, except by the two original portals of knowledge, sensation and reflection. If therefore any of these have as yet no being in the understanding, it is impossible by words or description to excite them there. A man who had never felt the sensation of heat, could not be brought to comprehend that sensation by any thing which we might say to explain it. If we would really produce the idea in him, it must be by applying the proper object to his senses, and bringing him within the influence of a heated body. When this is done, and experience has taught him the sensation to which men have annexed the name of heat, it then becomes to him the sign of that idea, and thenceforth understands the meaning of the term, which, before, all the words in the world would not have been sufficient to convey into his mind. The case is the same in respect of light and colours. A man born blind, and thereby deprived of the only means of acquiring the ideas of this class, can never be brought to understand the names by which they are expressed. The reason is obvious. They stand for ideas which have no existence in his mind; and as the organ appropriated to their reception is wanting, all other contrivances are vain, nor can they by any description be suggested to his imagination. But it is quite otherwise in the case of complex notions; for these being no more than certain combinations of simple ideas, put together in various forms, if the original ideas out of which the combinations are made have already obtained admission into the understanding, and the names serving to express them are known, it will be easy, by enumerating the several ideas concerned in the composition, and marking the order and manner in which they are united, to raise any complex conception in the mind. Thus the idea answering to the word rainbow may be readily excited in the imagination of another who has never seen the appearance itself, by barely describing the figure, size, position, and order of colours, if we suppose that these several simple ideas, with their names, are sufficiently known to him.

4. And this leads to a second observation upon this subject, namely, that words standing for complex ideas are all complex definable, whereas those by which we denote simple ideas are not; for simple ideas being a sort of secondary perceptions, which have no other entrance into the mind than by sensation or reflection, can only be obtained by experience, from the several objects of nature which are proper to produce in us those perceptions. Words indeed may very well serve to remind us of them, if they have already found admission into the understanding, and their connection with the established names be known; but they can never give them their original being and existence there. And hence it is, that when any one asks the meaning of a word denoting a simple idea, we pretend not to explain it to him by a definition, well knowing that to be impossible; but, Of Percep-supposing him already acquainted with the idea, and only ignorant of the name by which it is called, we either mention it to him by some other name with which we presume he knows its connection, or appeal to the object from which the idea itself is derived. Thus, were any one to ask the meaning of the word white, we should tell him it stood for the same idea as albus in Latin, or blanc in French; or, if we thought him a stranger to these languages, we might appeal to an object producing the idea, by saying that it denotes the colour which we observe in snow or milk. But this is by no means a definition of the word, exciting a new idea in his understanding; it is merely a contrivance to remind him of a known idea, and teach him its connection with the established name. For if the ideas after which he inquires have never yet been raised in his mind, as suppose one who had seen no other colours than black and white, should ask the meaning of the word scarlet, it is easy to perceive, that it would be no more possible to make him comprehend it by words, or by a definition, than to introduce the same perception into the imagination of a man born blind.

The only method in this case is, to present some object, by looking at which the perception itself may be excited; and thus he will learn both the name and the idea together.

5. But how comes it to pass that men agree in the names of their simple ideas, seeing they cannot ascertain the perceptions in one another's minds, nor by words make known these perceptions to others? The effect is produced by experience and observation. Thus finding, for instance, that the name of heat is annexed to that sensation which men feel when they approach the fire, we make it also the sign of the sensation excited in us by such an approach, nor have we any doubt that it denotes the same perception in our mind as in theirs. For we are naturally led to imagine, that the same objects operate alike upon the organs of the human body, and produce an uniformity of sensations. No man fancies that the idea raised in him by the taste of sugar, and which he calls sweetness, differs from that excited in another by the same object; or that wormwood, to the relish of which he has applied the epithet bitter, produces in another the sensation which he denotes by the word sweet. Presuming therefore upon this conformity of perceptions, when they arise from the same objects, we easily agree as to the names of our simple ideas; and if at any time, by a more narrow scrutiny into things, new ideas of this class should come in our way, which we may choose to express by terms of our own invention, these names are explained, not by a definition, but by referring to the objects whence the ideas themselves have been obtained.

6. Being in this manner furnished with simple ideas, and the names by which they are expressed, the meaning of the terms which stand for complex ideas is easily got at, because the ideas themselves answering to these terms may be conveyed into the mind by definitions; for our complex notions are only certain combinations of simple ideas. When, therefore, these are enumerated, and the manner in which they are united into one conception explained, nothing more is wanting to raise that conception in the understanding; and thus the term denoting it comes of course to be understood. But here it is worth while to reflect a little upon the wise contrivance of nature, in thus furnishing us with the aptest means of communicating our thoughts to others. For were it not so ordered, that we could thus by definitions convey our complex ideas from one to another, it would in many cases be impossible to make them known at all. This is apparent in those ideas which are the proper work of the mind itself; for as they exist only in the understanding, and have no real objects in nature in conformity to which they are framed, if we could not make them known by description, they would for ever lie hid within our own breasts, and be confined to the narrow acquaintance of a single mind. All the fine scenes which arise from time to time in the poet's fancy, and by his lively painting give such entertainment to his readers, were he destitute of this faculty of laying them open to the view of others by words and description, could not extend their influence beyond his own imagination, or give pleasure to any one but the original inventor.

7. There is this further advantage in the ability we enjoy of communicating our complex notions by definitions, that as these constitute by far the largest class of our ideas, and most frequently occur in the progress and improvement of knowledge, so they are by this means imparted with the greatest readiness, than which nothing can tend more to the increase and diffusion of science; for a definition is soon pursued, and if the terms of it are well understood, the idea itself finds an easy admission into the mind. But, in simple perceptions, where we are referred to the objects producing them, if these cannot be come at, as is sometimes the case, the names by which they are expressed must remain empty sounds. As new ideas of this class, however, occur but rarely in the sciences, they seldom create any great obstruction. Yet it is otherwise with our complex notions; for, every step we take leading us into new combinations and views of things, it becomes necessary to explain these to others, before they can be made acquainted with our discoveries; and as the manner of definitions is easy, requiring no apparatus but that of words, which are always ready at hand, we can hence with less difficulty remove such obstacles as might arise from terms of our own invention, when these are made to stand for new complex ideas suggested to the mind by some actual train of thinking. And thus at last we are let into the mystery hinted at in the beginning of this chapter, viz. how we may become acquainted with the thoughts of another, when he makes use of words to which we have as yet joined no ideas. From what has already been said, the answer is obvious. If the terms denote simple perceptions, he must refer us to those objects of nature whence the perceptions themselves are to be obtained; but if they stand for complex ideas, their meaning may be explained by a definition.

2. Of Definition.

1. A definition is the development and determination of some conception of the mind, answering to the word or defined term made use of as its sign. Now as, in exhibiting any idea to another, it is necessary that the description be such as may excite that precise idea in his mind, it is plain that definitions, properly speaking, are not arbitrary, but confined to the representing of certain determinate settled notions, such, namely, as are annexed by the speaker or writer to the words he employs. As nevertheless it is universally allowed that the signification of words is perfectly voluntary, and not the effect of any natural and necessary connection between them and the ideas for which they stand, some may perhaps wonder why definitions are not so likewise. In order, therefore, to unravel this difficulty, and show distinctly what is and what is not arbitrary in speech, we must carefully distinguish between the connection of our words and ideas, and the unfolding of the ideas themselves.

2. First, as to the connection of words and ideas; this, it is evident, is purely an arbitrary institution. When, for instance, we have in our minds the idea of any particular species of metals, the calling it by the name gold, is an effect of the voluntary choice of men speaking the same language, and not of any peculiar aptness in that sound to express that idea. Other nations we find make use of different sounds, and with the same effect. Thus Of Percep. aurum denotes the same idea in Latin, and or in French; and even the word gold itself would have as well served to express the idea of that metal which we call silver, had custom in the beginning so established it.

3. But although we are thus entirely at liberty to connect any idea with any sound, yet it is quite otherwise in unfolding the ideas themselves; for every idea having a precise appearance of its own, by which it is distinguished from every other idea, it is manifest, that in laying it open to others, we must study such a description as shall exhibit that peculiar appearance. When we have formed to ourselves the idea of a figure bounded by four equal sides, joined together at right angles, we are at liberty to express that idea by any sound, and call it either a square or a triangle. But whichever of these names we use, so long as the idea is the same, the description by which we would signify it to another must likewise remain so. Let it be called square or triangle, it is still a figure having four equal sides, and all its angles right ones. Hence we clearly see what is and what is not arbitrary in the use of words. The establishing any sound as the mark of some determinate idea in the mind, is the effect of free choice, and of a voluntary combination amongst men; and as different nations make use of different sounds to denote the same ideas, hence proceed all that variety of languages which we meet with in the world. But when a connection between our ideas and words is once settled, the unfolding of the idea answering to any word, which properly constitutes a definition, is by no means an arbitrary thing. In this case we are bound to exhibit the precise conception which either the use of language, or our own particular choice, has annexed to the term we employ.

4. And thus it appears, that definitions, considered as descriptions of ideas in the mind, are steady and invariable, being confined to the representation of those precise ideas. But then, in the application of definitions to particular names, we are altogether left to our own free choice; because, as the connecting of any idea with any sound is a perfectly arbitrary institution, the applying the description of that idea to that sound must be so likewise. When, therefore, logicians tell us that the definition of the name is arbitrary, they mean no more than this, That as different ideas may be connected with any term, according to the pleasure of him who employs it, in like manner may different descriptions be applied to the term, suitable to the ideas so connected. But this connection being settled, and the term considered as the sign of some fixed idea in the understanding, we are no longer left to arbitrary explanations, but must study such a description as corresponds with that precise idea. Now, this alone, according to what has been before laid down, ought to be accounted a definition. What seems to have occasioned so small confusion in this matter, is, that many explanations of words, where no idea is unfolded, but the connection between some word and idea is merely asserted, have yet been dignified with the name of definitions. Thus, when we say that a clock is an instrument by which we measure time, that is by some called a definition; and yet it is plain that we are beforehand supposed to have an idea of this instrument, and only taught that the word clock serves in common language to denote that idea. By this rule all explanations of words in our dictionaries would be definitions, nay, the names of even simple ideas might be thus described. White, we may say, is the colour we observe in snow or milk; heat, the sensation produced by approaching the fire; and so in innumerable other instances. But these, and all others of a similar kind, are by no means definitions, exciting new ideas in the understanding, but merely contrivances to remind us of known ideas, and to teach their connection with the established names.

5. But, in definitions properly so called, we have first to consider the term we use as the sign of some inward conception, either annexed to it by custom, or by our own free choice; and then the business of the definition is to unfold and explicate that idea. As therefore the whole art consists in giving just and true copies of our ideas, a definition is said to be made perfect, when it serves distinctly to excite the idea described in the mind of another, even supposing him to have been previously unacquainted with it. This point being settled, then, let us next inquire what those ideas are which are capable of being thus unfolded. And, in the first place, it is evident that all our simple ideas are necessarily excluded. We have already seen that experience alone is to be consulted here, insomuch that if either the objects whence they are derived come not in our way, or the avenues appointed by nature for their reception are wanting, no description can be sufficient to convey them into the mind. But where the understanding is already supplied with these original and primitive conceptions, as they may be united together in an infinity of different forms, so may all their several combinations be distinctly laid open, by enumerating the simple ideas concerned in the various combinations, and tracing the order and manner in which they are linked together in composition. Now these combinations of simple notices constitute what we call complex notions; whence it is evident, that complex ideas, and these alone, admit of that kind of description which goes by the name of definition.

6. Definitions, then, are pictures or representations of our ideas; and as these representations are only possible when the ideas themselves are complex, it is obvious to remark, that definitions cannot have place, except where we make use of terms standing for such complex ideas. But our complex ideas being, as we have said, nothing more than different combinations of simple ideas, we know and comprehend them perfectly, when we know the several simple ideas of which they consist, and can so put them together in our minds as may be necessary towards the framing of that peculiar connection which gives every idea its distinct and proper appearance.

7. Two things are therefore required in every definition. The first is, that all the original ideas, out of which the complex one is formed, be distinctly enumerated; and the second is, that the order and manner of combining them into one conception be clearly explained. Where a definition has these requisites, nothing is wanting to its perfection; because every one who reads it and understands the terms, seeing at once what ideas he is to join together, and also in what manner, can at pleasure form in his own mind the complex conception answering to the term defined. Let us, for instance, suppose the word square to stand for that idea by which we represent to ourselves a figure whose sides subtend quadrants of a circumscribed circle. The parts of this idea are the sides bounding the figure. These must be four in number, and all equal amongst themselves, because they are each to subtend a fourth part of the same circle. But, besides these component parts, we must also take notice of the manner of putting them together, if we would exhibit the precise idea for which the word square here stands; for, four equal right lines, anyhow joined, will not subtend quadrants of a circumscribed circle. A figure with this property must have its sides standing also at right angles. Taking in this last consideration, therefore, respecting the manner of combining the parts, the idea is fully described, and the definition thereby rendered complete; for a figure bounded by four equal sides, joined together at right angles, has the property required, and is moreover the only right-lined figure to which that property belongs.

8. It will now be obvious to every one in what manner we ought to proceed, in order to arrive at just and adequate definitions. First, we are to take an exact view of... Perception, the idea to be described; trace it to its original principles; and mark the several simple perceptions which enter into the composition of it. Secondly, we are to consider the particular manner in which these elementary ideas are combined, in order to form that precise conception for which the term we make use of stands. When this is done, and the idea is wholly unravelled, we have nothing more to do than fairly to transcribe the appearance it makes to our own minds. Such a description, by distinctly exhibiting the order and number of our primitive conceptions, cannot fail to excite, at the same time, in the mind of every one who reads it, the complex idea resulting from them; and therefore it attains the true and proper end of a definition.

3. Of the Composition and Resolutions of our Ideas, and the Rules of Definition thence arising.

1. The rule laid down in the foregoing chapter is general, extending to all possible cases, and is indeed that to which alone we can have recourse where any doubt or difficulty may arise. It is not, however, necessary that we should practise it in every particular instance. Many of our ideas are extremely complicated, insomuch that to enumerate all the simple perceptions out of which they are formed would be a very troublesome and tedious task. For this reason logicians have established certain copious rules of defining, which it may not be amiss here to give some account of. But in order to the better understanding of what follows, it will be necessary to keep in view, that there is a certain gradation in the composition of our ideas. The mind of man is very limited in its apprehension, and cannot take in a great number of objects at once. We are therefore obliged to proceed gradually, and make our first advances subservient to those which follow. Thus, in forming our complex notions, we begin at first with but a few simple ideas, such as we can manage with ease, and unite together into one conception. When we are provided with a sufficient stock of these, and have by habit and use rendered them familiar to our minds, they become the component parts of other ideas still more complicated, and form what may be called a second order of compound notions. This process, it is evident, may be continued to any degree of composition we please, mounting from one stage to another, and enlarging, as we ascend, the number of combinations.

2. But, in a series of this kind, whosoever would make himself perfectly acquainted with the last and highest order of ideas, must find it expedient to proceed gradually through all the intermediate steps. For, were he to take any very composite idea to pieces, and, without regard to the several classes of simple perceptions which have already been formed into distinct combinations, break it down at once into its original elements, the number would be so great as perfectly to confound the imagination, and overcome the utmost reach and capacity of the mind. When we observe a prodigious multitude of men huddled together in crowds, without order or any regular position, we find it impossible to arrive at an exact knowledge of their number. But if they are formed into separate battalions, and so stationed as to fall within the leisurely survey of the eye, by viewing them successively and in order, we may come to an easy and certain determination. It is the same in regard to complex ideas. When the original perceptions out of which they are framed are very numerous, it is not enough that we take a view of them in loose and scattered bodies; we must form them into distinct classes, and unite these classes in a just and orderly manner, before we can arrive at a true knowledge of the compound notions resulting from them.

3. This gradual progress of the mind to its composite notions, through a variety of intermediate steps, plainly points out the manner of conducting the definitions by which these notions may be conveyed into the minds of others; for, as the series commences with simple and easy combinations, and advances through a succession of different orders, rising one above another in the degree of composition, it is evident that, in a train of definitions expressing these ideas, a similar gradation is to be observed. Thus the complex ideas of the lowest order can no otherwise be described than by enumerating the simple ideas out of which they are formed, and explaining the manner of their union. But, in the second, or any other succeeding order, as they are formed out of those gradual combinations, it is not necessary, in describing them, to mention one by one all the simple ideas of which they are composed. They may be more distinctly and briefly unfolded by enumerating the compound ideas of a lower order, of whose union they are the result, and which are all supposed to be already known in consequence of previous definitions. Here then it is that the logical method of definition takes place; which, that it may be the better understood, we shall explain somewhat more particularly the several steps and gradations of the mind in compounding its ideas, and thence deduce that peculiar form of definition which logicians have thought proper to establish.

4. All the ideas we receive from the several objects of nature that surround us represent distinct individuals; and from these individuals, when compared together, are found in general to resemble each other. Hence, by collecting the resembling particulars into one conception, we form the notion of a species. And here let it be observed, that this last idea is less complicated than that by which we represent any of the particular objects contained under it; for the idea of the species excludes the peculiarities of the several individuals, and embraces such properties only as are common to all. Again, by comparing several species together, and observing their resemblance, we form the idea of a genus; where, in the same manner as before, the composition is lessened, because we leave out what is peculiar to the several species compared, and retain only the particulars in which they agree. It is easy to conceive the mind proceeding thus from one step to another, and advancing through several classes of general notions, until at last it arrives at the highest genus of all, denoted by the word being, where the idea of existence alone is concerned.

5. In this procedure we observe the mind unravelling a complex idea, and tracing it in the ascending scale, from the mind greater or less degrees of composition, until it terminates in one simple perception. But if we take the series the contrary way, and, beginning with the last or highest genus, carry our view downwards, through all the inferior genera and species, to the individuals, we shall thereby arrive at a distinct apprehension of the conduct of the understanding in the composition of its ideas; for, in the several classes of our perceptions, the highest in the scale is for the most part that which is made up of but a few simple ideas, such as the mind can take in and survey with ease. This first general notion, when branched out into the different subdivisions contained under it, has in every one of them something peculiar, by which they are distinguished amongst themselves; insomuch that, in descending from the genus to the species, we always superadd some new idea, and thereby increase the degree of composition. Thus the idea denoted by the word figure is of a very general nature, and composed of but few simple perceptions, as implying no more than space everywhere bounded. But if we descend further, and consider the boundaries of this space, as that they may be either lines or surface, we shall fall into the several species of figure. For where the space is bounded by one or more surfaces, we Of Judgment.

2. Experience.

4. The second ground of human judgment is Experience; from which we are led to infer the existence of those objects that surround us, and fall under the immediate notice of our senses. When we see the sun, or cast our eyes towards a building, we not only have perceptions of these objects within ourselves, but ascribe to them a real existence independent of the percipient mind. It is also by the information of the senses that we judge of the qualities of bodies; as when we say that "snow is white, fire hot, or steel hard." For as we are wholly unacquainted with the internal structure and constitution of the bodies which produce these sensations in us, and are unable to trace any connection between that structure and the sensations themselves, it is evident that we build our judgments altogether upon observation, ascribing to bodies such qualities as are answerable to the perceptions which they excite in us. Not that we ever suppose the qualities of bodies to be things of the same nature with our perceptions; for there is nothing in fire similar to the sensation of heat, or in a sword similar to that of pain; but when different bodies excite in our minds similar perceptions, we necessarily ascribe to these bodies, not only an existence independent of us, but likewise similar qualities, of which it is the nature to produce similar perceptions in the human mind. But this is not the only advantage derived from experience; for to that too we are indebted for all our knowledge regarding the co-existence of sensible qualities in objects, and the operations of bodies one upon another. Ivory, for instance, is hard and elastic; this we know by experience, and indeed by experience alone. For, being altogether strangers to the true nature both of elasticity and hardness, we cannot by the bare contemplation of our ideas determine how far the one necessarily implies the other, or whether there may not be a repugnance between them. But when we observe that both exist in the same object, we are then assured from experience that they are not incompatible; and when we also find that a stone is hard and not elastic, and that air though elastic is not hard, we likewise conclude upon the same foundation that the ideas are not necessarily conjoined, but may exist separately in different objects. In like manner, with regard to the operations of bodies upon one another, it is evident that our knowledge this way is all derived from observation. Aqua regia dissolves gold, as has been found by frequent trial, nor is there any other way of arriving at the discovery. Naturalists may tell us, if they please, that the parts of aqua regia are of a texture apt to insinuate between the corpuscles of gold, and thereby loosen and burst them asunder. If this be a true account of the matter, it will notwithstanding be allowed that our conjecture in regard to the conformation of these bodies is deduced from the experiment, and not the experiment from the conjecture. It was not from any previous knowledge of the intimate structure of aqua regia and gold, and the aptness of their parts to act or to be acted upon, that we came by the conclusion above mentioned. The internal constitution of bodies is in a manner wholly unknown to us; and could we even surmount this difficulty, yet, as the separation of the parts of gold implies something like an active force in the menstruum, and we are unable to conceive how the latter comes to be possessed of this activity, the effect must be owned to be altogether beyond our comprehension. But when repeated trials had once confirmed it, insomuch that it was admitted as an established truth in natural knowledge, it then became easy for men to spin out theories of their own invention, and contrive such a structure of parts, both for gold and aqua regia, as would best serve to explain the phenomenon upon the principles of that system of philosophy, whatever it might be, which they had adopted.

5. From what has been said, it is evident that as intuition is the foundation of what we call scientific, so experience is the foundation of what we call natural knowledge. For this last being wholly concerned with objects of sense, or those bodies that constitute the natural world, and their properties, as far as we can discover them, being to be traced only by a long and painful series of observations, it is apparent, that, in order to improve this branch of knowledge, we must betake ourselves to the method of trial and experiment.

6. But though experience is what we may term the immediate foundation of natural knowledge, yet with respect to particular persons its influence is very narrow and confined. The bodies that surround us are numerous, many of them lie at a great distance, and some are quite beyond our reach. Life is so short, and so crowded with cares, that but little time is left for any single man to employ himself in unfolding the mysteries of nature. Hence it is necessary to admit many things upon the testimony of others, which by this means becomes the foundation of a great part of our knowledge of body. No man doubts of the power of aqua regia to dissolve gold, though perhaps he never himself made the experiment. In these, therefore, and such like cases, we judge of the facts and operations of nature upon the mere ground of testimony. However, as we can always have recourse to experiment where any doubt or scruple arises, this is justly considered as the true foundation of natural philosophy, being indeed the ultimate support upon which our assent rests, and to which we appeal when the highest degree of evidence is required.

7. But there are many facts which do not admit of an appeal to the senses; and in this case testimony is the true and only foundation of our judgments. All human actions, of whatever kind, when considered as already past, are of the nature here described; because having now no longer any existence, both the facts themselves, and the circumstances attending them, can be known only from the relations of those who had sufficient opportunities of arriving at the truth. Testimony, therefore, is justly accounted a third ground of human judgment; and as from the other two we have deduced scientific and natural knowledge, so we may from this derive historical, by which we mean, not merely a knowledge of the civil transactions of states and kingdoms, but of all facts whatsoever, where testimony is the ultimate foundation of our belief.

2. Of Affirmative and Negative Propositions.

1. Whilst the comparing of our ideas is considered merely as an act of the mind, assembling them together, and joining or disjoining them according to the result of its perceptions, we call it judgment; but when our judgments are put into words, they then bear the name of propositions. A proposition, therefore, is a sentence expressing some judgment of the mind, by which two or more ideas are affirmed to agree or disagree. Now, as our judgments include at least two ideas, one of which is affirmed or denied of the other, so must a proposition have terms answering to these ideas. The idea of which we affirm or deny, and of course the term expressing that idea, is called the subject of the proposition. The idea affirmed or denied, as also the term answering it, is called the predicate. Thus, in the proposition, "God is omnipotent," God is the subject, it being of him that we affirm omnipotence, and om- nipotent is the predicate, because we affirm that the idea expressed by that word belongs to God.

2. But as in propositions ideas are either joined or disjoined, it is not enough to have terms expressing those ideas, unless we have also some words to denote their agreement or disagreement. That word in a proposition which connects two ideas together, is called the copula; and if a negative particle be annexed, we thereby understand that the ideas are disjoined. The substantive verb is commonly made use of for the copula, as in the above-mentioned proposition, "God is omnipotent," where is represents the copula, and signifies the agreement of the ideas of God and omnipotence. But if we mean to separate two ideas, then, besides the substantive verb, we must also use some particle of negation to express this repugnance. The proposition, "Man is not perfect," may serve as an example of this kind; for here the notion of perfection being removed from the idea of man, the negative particle not is inserted after the copula, to signify the disagreement between the subject and the predicate.

3. Every proposition necessarily consists of these three parts, but then it is not alike needless that they be all severally expressed in words; because the copula is often included in the term of the predicate, as when we say "He sits," which imports the same as "He is sitting." In the Latin language a single word has often the force of a whole sentence. Thus ambulat is the same as ille est ambulans; amo, as ego sum amans; and so in innumerable other instances, by which it appears that we are not so much to regard the number of words in a sentence, as the ideas which they represent, and the manner in which they are put together. For wherever two ideas are joined or disjoined in an expression, though of but a single word, it is evident that we have a subject, predicate, and copula, and of consequence a complete proposition.

4. When the mind joins two ideas, we call it an affirmative judgment; when it separates them, a negative; and as any two ideas compared together must necessarily either agree or not agree, it is evident that all our judgments fall under these two divisions. Hence, likewise, the propositions expressing these judgments are all either affirmative or negative. An affirmative proposition connects the predicate with the subject, as, "A stone is heavy;" a negative proposition separates them, as, "God is not the author of evil."

Affirmation, therefore, is the same as joining two ideas together; and this is effected by means of the copula. Negation, on the contrary, marks the repugnance between the ideas compared, in which case a negative particle must be called in to show that the connection included in the copula does not take place.

5. Hence we see the reason of the rule commonly laid down by logicians, That in all negative propositions the negation ought to affect the copula. For as the copula, when placed by itself between the subject and the predicate, manifestly binds them together, it is evident, that in order to render a proposition negative, the particles of negation must enter it in such a manner as to destroy this union. In a word, two ideas are only disjoined in a proposition when the negative particle may be so referred to the copula as to break the affirmation included in it, and undo that connection it would otherwise establish. When we say, for instance, "No man is perfect," take away the negation, and the copula of itself plainly unites the ideas in the proposition. But as this is the very reverse of what is intended, a negative mark is added, to show that this union does not take place here. The negation, therefore, by destroying the effect of the copula, changes the very nature of the proposition, insomuch that, instead of binding two ideas together, it denotes their separation. On the contrary, in this sentence, "The man who departs not from an upright behaviour is beloved of God," the predicate beloved of God is evidently affirmed of the subject an upright man; so that, notwithstanding the negative particle, the proposition is still affirmative. The reason is plain. The negation here affects not the copula, but, constituting properly a part of the subject, serves, with other terms in the sentence, to form one complex idea, of which the predicate beloved of God is directly affirmed.

3. Of Universal and Particular Propositions.

1. The next considerable division of propositions is that Division into universal and particular. Our ideas, according to what has been already observed, are all singular as they enter the mind, and represent individual objects. But as by abstraction we can render them universal, so as to comprehend a whole class of things, and sometimes several classes at once, hence the terms expressing these ideas must be in like manner universal. If, therefore, we suppose any general term to become the subject of a proposition, it is evident, that whatever is affirmed of the abstract idea belonging to that term, may be affirmed of all the individuals to which that idea extends. Thus, when we say, "Men are mortal," we consider mortality, not as confined to one or any number of particular men, but as what may be affirmed without restriction of the whole species. By this means the proposition becomes as general as the idea which forms the subject of it, and indeed derives its universality entirely from that idea, being more or less so according as this may be extended to more or fewer individuals. But it is further to be observed of these general terms, that they sometimes enter a proposition in their full latitude, as in the example above given, and sometimes appear with a mark of limitation. In this last case we are given to understand that the predicate agrees not to the whole universal idea, but only to a part of it; as in the proposition, "Some men are wise." Here wisdom is not affirmed of every particular man, but is restricted to a few of the human species.

2. Now, from this different appearance of the general Proposition-idea which constitutes the subject of any judgment, arises the division of propositions into universal and particular. Universal an universal proposition is that in which the subject is some general term taken in its full latitude, insomuch that the predicate agrees to all the individuals comprehended under it if it denotes a proper species, and to all the several species and their individuals if it marks an idea of a higher order. The words all, every, no, none, &c. are the proper signs of this universality; and as they seldom fail to accompany general truths, so they form the most obvious criterion by which to distinguish them. "All animals have a power of beginning motion." This is an universal proposition, as we know from the word all prefixed to the subject animals, which denotes that it must be taken in its full extent. Hence the power of beginning motion may be affirmed of all the different species of animals.

3. A particular proposition has in like manner some Particular general term for its subject, but with a mark of limitation added, to denote that the predicate agrees only to some of the individuals comprehended under a species, or to one

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1 See the preceding note, where it is demonstrated that the terms alone, and not the ideas, are in reality general. The term man is equally applicable to every individual of the human race; and therefore, what is affirmed or denied of men in general, is affirmed or denied of all the individuals, without regard to their discriminating qualities. Some is a definitive word (see Grammar), which, prefixed to the word man, limits the signification of that general term; and therefore what is affirmed of some men, is affirmed only of part of the race, but that part itself is not ascertained. mutable truths. If it be true that the whole is equal to all its parts, it must be so unchangeably; because the relation of equality, being attached to the ideas themselves, must ever intervene where the same ideas are compared. Of this nature are all the truths of natural religion, morality, and mathematics, and, in general, whatever may be gathered from the bare view and consideration of our ideas.

4. The second ground of human judgment is Experience; from which we are led to infer the existence of those objects that surround us, and fall under the immediate notice of our senses. When we see the sun, or cast our eyes towards a building, we not only have perceptions of these objects within ourselves, but ascribe to them a real existence independent of the percipient mind. It is also by the information of the senses that we judge of the qualities of bodies; as when we say that "snow is white, fire hot, or steel hard." For as we are wholly unacquainted with the internal structure and constitution of the bodies which produce these sensations in us, and are unable to trace any connection between that structure and the sensations themselves, it is evident that we build our judgments altogether upon observation, ascribing to bodies such qualities as are answerable to the perceptions which they excite in us. Not that we ever suppose the qualities of bodies to be things of the same nature with our perceptions; for there is nothing in fire similar to the sensation of heat, or in a sword similar to that of pain; but when different bodies excite in our minds similar perceptions, we necessarily ascribe to these bodies, not only an existence independent of us, but likewise similar qualities, of which it is the nature to produce similar perceptions in the human mind. But this is not the only advantage derived from experience; for to that too we are indebted for all our knowledge regarding the co-existence of sensible qualities in objects, and the operations of bodies one upon another. Ivory, for instance, is hard and elastic; this we know by experience, and indeed by experience alone. For, being altogether strangers to the true nature both of elasticity and hardness, we cannot by the bare contemplation of our ideas determine how far the one necessarily implies the other, or whether there may not be a repugnance between them. But when we observe that both exist in the same object, we are then assured from experience that they are not incompatible; and when we also find that a stone is hard and not elastic, and that air though elastic is not hard, we likewise conclude upon the same foundation that the ideas are not necessarily conjoined, but may exist separately in different objects. In like manner, with regard to the operations of bodies upon one another, it is evident that our knowledge this way is all derived from observation. Aqua regia dissolves gold, as has been found by frequent trial, nor is there any other way of arriving at the discovery. Naturalists may tell us, if they please, that the parts of aqua regia are of a texture apt to insinuate between the corpuscles of gold, and thereby loosen and burst them asunder. If this be a true account of the matter, it will notwithstanding be allowed that our conjecture in regard to the conformation of these bodies is deduced from the experiment, and not the experiment from the conjecture. It was not from any previous knowledge of the intimate structure of aqua regia and gold, and the aptness of their parts to act or to be acted upon, that we came by the conclusion above mentioned. The internal constitution of bodies is in a manner wholly unknown to us; and could we even surmount this difficulty, yet, as the separation of the parts of gold implies something like an active force in the menstruum, and we are unable to conceive how the latter comes to be possessed of this activity, the effect must be owned to be altogether beyond our comprehension. But when repeated trials had once confirmed it, insomuch that it was admitted as an established truth in natural knowledge, it then became easy for men to spin out theories of their own invention, and contrive such a structure of parts, both for gold and aqua regia, as would best serve to explain the phenomenon upon the principles of that system of philosophy, whatever it might be, which they had adopted.

5. From what has been said, it is evident that as intuition is the foundation of what we call scientific, so experience is the foundation of what we call natural knowledge. For this last being wholly concerned with objects of sense, or those bodies that constitute the natural world, and their properties, as far as we can discover them, being to be traced only by a long and painful series of observations, it is apparent, that, in order to improve this branch of knowledge, we must betake ourselves to the method of trial and experiment.

6. But though experience is what we may term the immediate foundation of natural knowledge, yet with respect to particular persons its influence is very narrow and confined. The bodies that surround us are numerous, many of them lie at a great distance, and some are quite beyond our reach. Life is so short, and so crowded with cares, that but little time is left for any single man to employ himself in unfolding the mysteries of nature. Hence it is necessary to admit many things upon the testimony of others, which by this means becomes the foundation of a great part of our knowledge of body. No man doubts of the power of aqua regia to dissolve gold, though perhaps he never himself made the experiment. In these, therefore, and such like cases, we judge of the facts and operations of nature upon the mere ground of testimony. However, as we can always have recourse to experiment where any doubt or scruple arises, this is justly considered as the true foundation of natural philosophy, being indeed the ultimate support upon which our assent rests, and to which we appeal when the highest degree of evidence is required.

7. But there are many facts which do not admit of an appeal to the senses; and in this case testimony is the true and only foundation of our judgments. All human actions, of whatever kind, when considered as already past, are of the nature here described; because having now no longer any existence, both the facts themselves, and the circumstances attending them, can be known only from the relations of those who had sufficient opportunities of arriving at the truth. Testimony, therefore, is justly accounted a third ground of human judgment; and as from the other two we have deduced scientific and natural knowledge, so we may from this derive historical, by which we mean, not merely a knowledge of the civil transactions of states and kingdoms, but of all facts whatsoever, where testimony is the ultimate foundation of our belief.

2. Of Affirmative and Negative Propositions.

1. Whilst the comparing of our ideas is considered merely as an act of the mind, assembling them together, and joining or disjoining them according to the result of its previous perceptions, we call it judgment; but when our judgments are put into words, they then bear the name of propositions. A proposition, therefore, is a sentence expressing some judgment of the mind, by which two or more ideas are affirmed to agree or disagree. Now, as our judgments include at least two ideas, one of which is affirmed or denied of the other, so must a proposition have terms answering to these ideas. The idea of which we affirm or deny, and of course the term expressing that idea, is called the subject of the proposition. The idea affirmed or denied, as also the term answering it, is called the predicate. Thus, in the proposition, "God is omnipotent," God is the subject, it being of him that we affirm omnipotence, and om- nipotent is the predicate, because we affirm that the idea expressed by that word belongs to God.

2. But as in propositions ideas are either joined or disjoined, it is not enough to have terms expressing those ideas, unless we have also some words to denote their agreement or disagreement. That word in a proposition which connects two ideas together, is called the copula; and if a negative particle be annexed, we thereby understand that the ideas are disjoined. The substantive verb is commonly made use of for the copula, as in the above-mentioned proposition, "God is omnipotent," where is represents the copula, and signifies the agreement of the ideas of God and omnipotence. But if we mean to separate two ideas, then, besides the substantive verb, we must also use some particle of negation to express this repugnance. The proposition, "Man is not perfect," may serve as an example of this kind; for here the notion of perfection being removed from the idea of man, the negative particle not is inserted after the copula, to signify the disagreement between the subject and the predicate.

3. Every proposition necessarily consists of these three parts, but then it is not alike needful that they be all severally expressed in words; because the copula is often included in the term of the predicate, as when we say "He sits," which imports the same as "He is sitting." In the Latin language a single word has often the force of a whole sentence. Thus ambulat is the same as ille est ambulans; amo, as ego sum amans; and so in innumerable other instances, by which it appears that we are not so much to regard the number of words in a sentence, as the ideas which they represent, and the manner in which they are put together. For wherever two ideas are joined or disjoined in an expression, though of but a single word, it is evident that we have a subject, predicate, and copula, and of consequence a complete proposition.

4. When the mind joins two ideas, we call it an affirmative judgment; when it separates them, a negative; and as any two ideas compared together must necessarily either agree or not agree, it is evident that all our judgments fall under these two divisions. Hence, likewise, the propositions expressing these judgments are all either affirmative or negative. An affirmative proposition connects the predicate with the subject, as, "A stone is heavy;" a negative proposition separates them, as, "God is not the author of evil." Affirmation, therefore, is the same as joining two ideas together; and this is effected by means of the copula. Negation, on the contrary, marks the repugnance between the ideas compared, in which case a negative particle must be called in to show that the connection included in the copula does not take place.

5. Hence we see the reason of the rule commonly laid down by logicians, That in all negative propositions the negation ought to affect the copula. For as the copula, when placed by itself between the subject and the predicate, manifestly binds them together, it is evident, that in order to render a proposition negative, the particles of negation must enter it in such a manner as to destroy this union. In a word, two ideas are only disjoined in a proposition when the negative particle may be so referred to the copula as to break the affirmation included in it, and undo that connection it would otherwise establish. When we say, for instance, "No man is perfect," take away the negation, and the copula of itself plainly unites the ideas in the proposition. But as this is the very reverse of what is intended, a negative mark is added, to show that this union does not take place here. The negation, therefore, by destroying the effect of the copula, changes the very nature of the proposition, insomuch that, instead of binding two ideas together, it denotes their separation. On the contrary, in this sentence, "The man who departs not from an upright behaviour is beloved of God," the predicate beloved of God is evidently affirmed of the subject an upright man; so that, notwithstanding the negative particle, the proposition is still affirmative. The reason is plain. The negation here affects not the copula, but, constituting properly a part of the subject, serves, with other terms in the sentence, to form one complex idea, of which the predicate beloved of God is directly affirmed.

3. Of Universal and Particular Propositions.

1. The next considerable division of propositions is that Division into universal and particular. Our ideas, according to what has been already observed, are all singular as they enter the mind, and represent individual objects. But as by abstraction we can render them universal, so as to comprehend a whole class of things, and sometimes several classes at once, hence the terms expressing these ideas must be in like manner universal. If, therefore, we suppose any general term to become the subject of a proposition, it is evident, that whatever is affirmed of the abstract idea belonging to that term, may be affirmed of all the individuals to which that idea extends. Thus, when we say, "Men are mortal," we consider mortality, not as confined to one or any number of particular men, but as what may be affirmed without restriction of the whole species. By this means the proposition becomes as general as the idea which forms the subject of it, and indeed derives its universality entirely from that idea, being more or less so according as this may be extended to more or fewer individuals. But it is further to be observed of these general terms, that they sometimes enter a proposition in their full latitude, as in the example above given, and sometimes appear with a mark of limitation. In this last case we are given to understand that the predicate agrees not to the whole universal idea, but only to a part of it; as in the proposition, "Some men are wise." Here wisdom is not affirmed of every particular man, but is restricted to a few of the human species.

2. Now, from this different appearance of the general Proposition idea which constitutes the subject of any judgment, arises the division of propositions into universal and particular. An universal proposition is that in which the subject is some general term taken in its full latitude, insomuch that the predicate agrees to all the individuals comprehended under it if it denotes a proper species, and to all the several species and their individuals if it marks an idea of a higher order. The words all, every, no, none, &c., are the proper signs of this universality; and as they seldom fail to accompany general truths, so they form the most obvious criterion by which to distinguish them. "All animals have a power of beginning motion." This is an universal proposition, as we know from the word all prefixed to the subject animals, which denotes that it must be taken in its full extent. Hence the power of beginning motion may be affirmed of all the different species of animals.

3. A particular proposition has in like manner some particular general term for its subject, but with a mark of limitation proposed, to denote that the predicate agrees only to some of the individuals comprehended under a species, or to one

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1 See the preceding note, where it is demonstrated that the terms alone, and not the ideas, are in reality general. The term man is equally applicable to every individual of the human race; and therefore, what is affirmed or denied of men in general, is affirmed or denied of all the individuals, without regard to their discriminating qualities. Some is a definitive word (see Grammar), which, prefixed to the word man, limits the signification of that general term; and therefore what is affirmed of some men, is affirmed only of part of the race, but that part itself is not ascertained. or more of the species belonging to a genus, and not to the entire universal idea. Thus, "Some stones are heavier than iron;" "Some men have an uncommon share of prudence." In the latter of these propositions, the subject some men implies only a certain number of individuals, comprehended under a single species. In the former, where the subject is a genus which extends to a great variety of distinct classes, some stones may not only imply any number of particular stones, but also several whole species of stones, inasmuch as there may be not a few with the property there described. Hence we see that a proposition does not cease to be particular by the predicate's agreeing to a whole species, unless that species, singly and distinctly considered, forms also the subject of which we affirm or deny.

4. There is still one species of propositions which remains to be described, and which the more deserves our notice, as it is not yet agreed amongst logicians to which of the two classes mentioned above they ought to be referred; namely, singular propositions, or those where the subject is an individual. Of this nature are the following: "Sir Isaac Newton was the inventor of fluxions;" "This book contains many useful truths." What occasions some difficulty as to the proper rank of these propositions is, that the subject being taken according to the whole of its extension, they sometimes have the same effect in reasoning as universals. But if it be considered that they are in truth the most limited kind of particular propositions, and that no proposition can with any propriety be called universal but where the subject is some universal idea, we shall not find any difficulty in determining to which class they ought to be referred. When we say, "Some books contain useful truths," the proposition is particular, because the general term appears with a mark of restriction. If, therefore, we say, "This book contains useful truths," it is evident that the proposition must be still more particular, as the limitation implied in the word this is of a more confined nature than in the former case.

5. We perceive, therefore, that all propositions are either affirmative or negative; nor is it less evident that in both cases they may be universal or particular. Hence arises that celebrated fourfold division of them into universal affirmative and universal negative, particular affirmative and particular negative, which indeed comprehends all their varieties. The use of this method of distinguishing them will appear more fully afterwards, when we come to treat of reasoning and syllogism.

4. Of Absolute and Conditional Propositions.

1. The objects about which we are chiefly conversant in this world, are all of a nature very liable to change. What may be affirmed of them at one time, cannot often be affirmed at another; and one of the objects of our knowledge is to enable us to distinguish rightly these variations, and trace the reasons upon which they depend. For it is observable, that amidst all the vicissitudes of nature, some things remain constant and invariable; nor are even the changes to which we see others liable, effected, except in consequence of uniform and steady laws, which, when known, are sufficient to direct us in our judgments concerning them. Hence philosophers, in distinguishing the objects of our perception into various classes, have been very careful to state, that some properties belong so essentially to the general idea, as not to be separable from it except by destroying its very nature; whilst others are only accidental, and may be affirmed or denied of it in different circumstances. Thus solidity, a yellow colour, and a great weight, are considered as essential qualities of gold; but whether it shall exist as an uniform conjoined mass, is not alike necessary. We find that by a proper menstrum it may be reduced to a very fine powder, and that an intense heat will bring it into a state of fusion.

2. From this diversity in the several qualities of things arises a considerable difference as to the manner of our judging concerning them. For all such properties as are inseparable from objects when considered as belonging to any genus or species, are affirmed absolutely and without reserve of that general idea. Thus we say, "Gold is very weighty;" "A stone is hard;" "Animals have a power of self-motion." But in the case of mutual or accidental qualities, as they depend upon some other consideration distinct from the general idea, that must also be taken into the account, in order to form an accurate judgment. Should we affirm, for instance, of some stones, that they are very susceptible of a rolling motion, the proposition, whilst it remains in this general form, cannot with any advantage be introduced into our reasonings. An aptness to receive that mode of motion flows from the figure of the stone, which, as it may vary infinitely, so our judgment only becomes applicable and determinate, when the particular figure, of which volatility is a consequence, is also taken into the account. Let us then introduce this other consideration, and the proposition will run as follows: "Stones of a spherical form are easily put into a rolling motion." Here we see the condition upon which the predicate is affirmed, and therefore know in what particular cases the proposition may be applied.

3. This consideration of propositions respecting the manner in which the predicate is affirmed of the subject, gives rise to the division of them into absolute and conditional. Absolute propositions are those in which we affirm some property inseparable from the idea of the subject, and which therefore belongs to it in all possible cases; as, "God is infinitely wise;" "Virtue tends to the ultimate happiness of man." But where the predicate is not necessarily connected with the idea of the subject, unless upon some consideration distinct from that idea, there the proposition is called conditional. The reason of the name is taken from the supposition annexed, which is of the nature of a condition, and may be expressed as such, thus: "If a stone be exposed to the rays of the sun, it will contract some degree of heat;" "If a river run in a very declining channel, its rapidity will constantly increase."

4. There is not any thing of greater importance in philosophy than due attention to this division of propositions, since if we be careful never to affirm things absolutely excepting where the ideas are inseparably conjoined, and if in our other judgments we distinctly mark the conditions which determine the predicate to belong to the subject, we shall be the less liable to mistake in applying general truths to the particular concerns of human life. It is owing to the exact observance of this rule that mathematicians have been so fortunate in their discoveries, and that what they demonstrate of magnitude in general may be applied with ease in all obvious occurrences.

5. The truth of it is, that particular propositions are only known to be true, when we can trace their connection with universals; and it is accordingly the great business of science to find out general truths which may be applied with safety in all obvious instances. Now the great advantage arising from determining with care the conditions upon which one idea may be affirmed or denied of another, is this; that thereby particular propositions really become universal, may be introduced with certainty into our reasonings, and serve as standards to conduct and regulate our judgments. To illustrate this by a familiar instance; if we say, "Some water acts very forcibly," the proposition is particular; and as the conditions on which this forcible action depends are not mentioned, it is as yet uncertain in what cases it may be applied. Let us then supply these conditions, and the proposition will run thus: "Water conveyed in sufficient quantity along a steep descent acts very forceibly." Here we have an universal judgment, inasmuch as the predicate forcible action may be ascribed to all water under the circumstances mentioned. Nor is it less evident that the proposition in this new form is of easy application; and in fact we find that men do apply it in instances where the forcible action of water is required, as in corn-mills and many other works of art.

5. Of Simple and Compound Propositions.

1. Hitherto we have treated of propositions where only two ideas are compared together. These are in the general called simple, because, having but one subject and one predicate, they are the effect of a simple judgment, that admits of no subdivision. But if it so happens that several ideas offer themselves to our thoughts at once, by which we are led to affirm the same thing of different objects, or different things of the same object, then the propositions expressing these judgments are called compound; because they may be resolved into as many others as there are subjects or predicates in the whole complex determination on the mind. Thus, "God is infinitely wise and infinitely powerful." Here there are two predicates, infinite wisdom, and infinite power, both affirmed of the same subject; and accordingly the proposition may be resolved into two others, affirming these predicates severally. In like manner, in the proposition, "Neither kings nor people are exempt from death," the predicate is denied of both subjects, and may therefore be separated from them in distinct propositions. Nor is it less evident, that if a complex judgment consists of several subjects and predicates, it may be resolved into as many simple propositions as are the number of different ideas compared together. "Riches and honours are apt to elate the mind, and increase the number of our desires." In this judgment there are two subjects and two predicates, and it is at the same time apparent that it may be resolved into four distinct propositions. "Riches are apt to elate the mind;" "Riches are apt to increase the number of our desires." And so of "honours."

2. Logicians have divided these compound propositions into a great many different classes; but, in our opinion, not with a due regard to their proper definition. Thus, conditionals, causals, relatives, and others, are mentioned as so many distinct species of this kind, though in fact they are no more than simple propositions. To give an instance of a conditional; "If a stone be exposed to the rays of the sun, it will contract some degree of heat." Here we have but one subject and one predicate; for the complex expression, a stone exposed to the rays of the sun, constitutes the proper subject of this proposition, and is no more than one determinate idea. The same thing happens in causals. Thus, "Rehoboam was unhappy because he followed evil counsel." Here there is an appearance of two propositions arising from the complexity of the expression; but when we come to consider the matter more nearly, it is evident that we have but a single subject and predicate. "The pursuit of evil counsel brought misery upon Rehoboam." It is not enough, therefore, to render a proposition compound, that the subject and predicate are complex notions, requiring sometimes a whole sentence to express them; for in this case the comparison is still confined to two ideas, and constitutes what we call a simple judgment. But where there are several subjects or predicates, or both, as the affirmation or negation may be alike extended to them all, the proposition expressing such a judgment is truly a collection of as many simple ones as there are different ideas compared. Confining ourselves, therefore, to this more strict and just notion of compound propositions, they are all reducible to two kinds, viz. copulatives and disjunctives.

3. A copulative proposition is, where the subjects and predicates are so linked together that they may be all severally affirmed or denied one of another. Of this nature are the examples of compound propositions given above. "Riches and honours are apt to elate the mind, and increase the number of our desires;" "Neither kings nor people are exempt from death." In the first of these, the two predicates may be affirmed severally of each subject, whence we have four distinct propositions. The other furnishes an example of the negative kind, where the same predicate, being disjoined from both subjects, may be also denied of them in separate propositions.

4. The other species of compound propositions are those called disjunctive; in which, comparing several predicates with the same subject, we affirm that one of them necessarily belongs to it, but leave the particular predicate undetermined. If any one, for example, says, "This world either exists of itself, or is the work of some all-wise and powerful cause," it is evident that one of the two predicates must belong to the world; but as the proposition determines not which, it is therefore of the kind we call disjunctive. Such too are the following: "The sun either moves round the earth, or is the centre about which the earth revolves;" "Friendship finds men equal, or makes them so." It is the nature of all propositions of this class, supposing them to be exact in point of form, that upon determining the particular predicate, the rest are of course to be removed; or if all the predicates but one are removed, that one necessarily takes effect, to the exclusion of the others. Thus, in the example given above, if we allow the world to be the work of some wise and powerful cause, we of course deny it to be self-existent; or if we deny it to be self-existent, we must necessarily admit that it was produced by some wise and powerful cause. Now this particular manner of linking the predicates together, so that the establishing of one displaces all the rest, or the excluding all but one necessarily establishes that one, cannot otherwise be effected than by means of disjunctive particles. And hence it is that propositions of this class take their names from these particles, which form so necessary a part of them, and indeed constitute their very nature considered as a distinct species.

6. Of the Division of Propositions into Self-evident and Demonstrable.

1. When any proposition is offered to the view of the mind, if the terms in which it is expressed be understood, upon comparing the ideas together, the agreement or dis-vided into agreement asserted will either be immediately perceived, or self-evident will be found to lie beyond the actual reach of the understanding. In the first case, the proposition is said to be self-evident, and admits not of any proof, because a bare attention to the ideas themselves produces full conviction and certainty; nor is it possible to call in any thing more evident by way of confirmation. But where the connection or repugnance comes not so readily under the inspection of the mind, we must have recourse to reasoning; and if by a clear series of proofs we can make out the truth proposed, insomuch that self-evidence shall accompany every step of the procedure, we are then able to demonstrate what we assert, and the proposition itself is said to be demonstrable. When we affirm, for instance, "That it is impossible for the same thing to be and not to be," whoever understands the terms made use of, perceives at the first glance the truth of what is asserted, nor can he by any efforts bring himself to believe the contrary. The proposition, therefore, is self-evident, and such that it is impossible by reasoning to make it plainer; because there is no truth more obvious or better known, from which it may be deduced as a consequence. But if we say, "This world had a beginning," the assertion is indeed equally true, but shines not forth with the same degree of evidence. We find great difficulty in con- ceiving how the world could be made out of nothing, and are not brought to a free and full consent, until by reasoning we arrive at a clear view of the absurdity involved in the contrary supposition. Hence this proposition is of the kind we call demonstrable, inasmuch as its truth is not immediately perceived by the mind, but yet may be made clear by means of others more known and obvious, whence it follows as an unavoidable consequence.

2. From what has been said, it appears that reasoning is employed only about demonstrable propositions, and that our intuitive and self-evident perceptions are the ultimate foundation upon which it rests.

3. Self-evident propositions furnish the first principles of reasoning; and it is certain, that if in our researches we employ only such principles as possess this character of self-evidence, and apply them according to the rules to be afterwards explained, we shall be in no danger of error in advancing from one discovery to another. For this we may appeal to the writings of the mathematicians, which being conducted by the express model here mentioned, furnish incontestable proof of the firmness and stability of human knowledge, when built upon so sure a foundation. For not only have the propositions of this science stood the test of ages, but they are found attended with that invincible evidence, which forces the assent of all who duly consider the proofs upon which they are established. Since the mathematicians are universally allowed to have hit upon the right method of arriving at unknown truths, since they have been the happiest in the choice as well as the application of their principles, it may not be amiss to explain here their method of stating self-evident propositions, and applying them to the purposes of demonstration.

4. First, then, it is to be observed, that they have been very careful in ascertaining their ideas, and fixing the significance of their terms. For this purpose they begin with definitions, in which the meaning of their words is so distinctly explained, that they cannot fail to excite in the mind of an attentive reader the very same ideas which are annexed to them by the writer. And indeed the clearness and irresistible evidence of mathematical knowledge is owing to nothing so much as this care in laying the foundation. Where the relation between any two ideas is accurately and justly traced, it will not be difficult for another to comprehend that relation, if in setting himself to discover it he brings the very same ideas into comparison. But if, on the contrary, he affixes to his words ideas different from those that were in the mind of him who first advanced the demonstration, it is evident, that as the same ideas are not compared, the same relation cannot subsist, insomuch that a proposition will be rejected as false, which, had the terms been rightly understood, must have appeared incontestably true. A square, for instance, is a figure bounded by four equal right lines, joined together at right angles. Here the nature of the angles constitutes a part of the idea no less than the equality of the sides; and many properties demonstrated of the square are deduced entirely from its being a rectangular figure. If therefore we suppose a man, who has formed a partial notion of a square, comprehending only the equality of its sides, without regard to the angles, reading some demonstration which implies also this latter consideration, it is plain that he would reject it as not universally true, inasmuch as it could not be applied where the sides were joined together at equal angles. For this last figure, answering still to his idea of a square, would be yet found without the property assigned to it in the proposition. But if he comes afterwards to correct his notion, and to render his idea complete, he will then readily admit the truth and justness of the demonstration.

5. We see, therefore, that nothing contributes so much to the improvement and certainty of human knowledge, as the having determinate ideas, and keeping them steady and invariable in all our discourses and reasonings concerning them. On this account it is that mathematicians, as has already been observed, always begin by defining their terms, and distinctly unfolding the notions which they are intended to express. Hence such as apply themselves to these studies have exactly the same views of things; and, bringing always the very same ideas into comparison, they readily discern the relations between them. It is likewise of importance, in every demonstration, to express the same idea invariably by the same word. From this practice mathematicians never deviate; and if it be necessary in their demonstrations, where the reader's comprehension is aided by a diagram, it is much more so in all reasonings concerning moral or intellectual truths, where the ideas cannot be represented by a diagram. The observation of this rule may sometimes be productive of ill-sounding periods; but when truth is the object, euphony ought to be disregarded.

6. When the mathematicians have taken this first step, and made known the ideas the relations of which they intend to investigate, their next care is, to lay down some self-evident truths, which may serve as a foundation for their future reasonings. And here indeed they proceed with remarkable circumspection, admitting no principles but such as flow immediately from their definitions, and necessarily force themselves upon every mind in any degree attentive to its ideas. Thus a circle is a figure formed by a right line moving round some fixed point in the same plane. The fixed point round which the line is supposed to move, and where one of its extremities terminates, is called the centre of the circle. The other extremity, which is conceived to be carried round until it returns to the point whence it set out, describes a curve running into itself, which is termed the circumference. All right lines drawn from the centre to the circumference are called radii. From this definition geometricians derive the self-evident truth, "That the radii of the same circle are all equal to one another;" or rather, perhaps, it is part of, and involved in, the very definition.

7. We may now observe, that in all propositions we either affirm or deny some property of the idea which constitutes the subject of our judgment, or we maintain that something may be done or effected. The first sort are called speculative or relative propositions, as in the example mentioned above. The others are called practical, for a reason too obvious to be mentioned; thus, "That a right line may be drawn from one point to another," is a practical proposition, inasmuch as it expresses that something may be done.

8. From this twofold consideration of propositions arises the twofold division of mathematical principles into axioms, postulates, and postulates. By an axiom is understood any self-evident speculative truth; such as, "That the whole is greater than its parts;" "That things equal to one and the same thing are equal to one another." But a self-evident practical proposition is what is denominated a postulate. Such are those of Euclid: "That a finite right line may be continued directly forwards;" "That a circle may be described about any centre with any distance." And here we have to observe, that as in an axiom the agreement or disagreement between the subject and predicate must come under the immediate inspection of the mind; so in a postulate, not only the possibility of the thing asserted must be evident at first view, but also the manner in which it may be effected. For where this manner is not of itself apparent, the proposition comes under the notion of the demonstrable kind, and is treated as such by geometrical writers. Thus, "To draw a right line from one point to another," is assumed by Euclid as a postulate, because the manner of doing it is so obvious, as to require no previous teaching. But then it is not equally evident how we are to construct an equilateral triangle. For this reason he advances it