LOGARITHMS.

LOGARITHMS, (from ratio, and number), the indices of the ratios of numbers to one another; being a series of numbers in arithmetical progression, corresponding to others in geometrical progression; by means of which, arithmetical calculations can be made with much more ease and expedition than otherwise.

SECT. I. History of Logarithms.

THE invention of logarithms first occurred to those versant in the construction of trigonometrical tables, in which immense labour was required by large multiplications, divisions, and extraction of roots. The aim proposed was, to reduce as much as possible the mul-

tiplications and divisions to additions and subtractions; and for this purpose, a method was invented by Nicholas Raymer Ursus Dithmarfus, which serves for one case of the sines, viz. when the radius is the first term in proportion, and the sines of two arcs the second and third terms. In this case the fourth term is found by only taking half the sum or difference of the sines of the other two arcs, and the complement of the greater. This method was first published in 1588, and a few years afterward was greatly improved by Clavius, who used it in all proportions in the solution of spherical triangles; adapting it to sines, tangents, versed sines, and secants; and this, whether the radius was the first term in the proportion or not.

This method, however, though now become much more

more generally useful than before, was still found attended with trouble in some cases; and as it depended upon certain properties of lines belonging to the circle, was rather of a geometrical than arithmetical nature; on which account the calculators about the end of the 16th and beginning of the 17th century, finding the solution of astronomical problems extremely troublesome, by reason of the tedious multiplications and divisions they required, continued their endeavours to lessen that labour, by searching for a method of reducing their operations to addition and subtraction. The first step towards this was, the consideration, that as in multiplication the ratio of the multiplier to unity is the same as that of the product to the multiplicand, it will follow, that the ratio of the product to unity must be equal to the sum of the two ratios of the multiplier to unity, and of the multiplicand to unity. Could a set of numbers therefore be found, which would represent the ratios of all other numbers to unity, the addition of two of the former set of numbers would be equivalent to the multiplication of the two numbers together, the ratios of which they denoted; and the sum arising from this addition would denote the ratio of their product to unity; whence the product itself might be found by looking for the corresponding natural number in the table.

The next thing was to fall upon a method of calculating such a table as was wanted, which indeed appeared an Herculean labour. The first observation was, that whatever numbers might be made use of to represent the ratios of others, the ratio of equality, or that of unity to unity must be 0; for that ratio has properly no magnitude, neither increasing nor diminishing any other ratio to which it is adapted, or from which it is subtracted.

2. The second observation was, that though any number might be chosen at pleasure to represent the ratio of any other number to unity, yet when once this choice was made, all the other numbers representing the different ratios must be determined by it. Thus, if the ratio of 10 to 1 be represented by 1, then the ratio of 100 to 1 must be 2, and that of 1000 to 1 must be 3, &c.; or if 2 was chosen to represent the ratio of 10 to 1, then that of 100 to 1 must be 4, that of 1000 to 1 must be 6, &c.; and no other numbers could possibly be used.

3. As those artificial numbers represented, or were proportional to, the ratios of the natural numbers to unity, they must be expressions of the numbers of some smaller equal ratios contained in the former and larger ones. Thus, if we make 1 the representative of the ratio of 10 to 1; then 3, which represents the ratio of 1000 to 1, will likewise express the number of ratios of 10 to 1, which are contained in that of 1000 to 1. If instead of 1, we make 1000 to be the ratio of 10 to 1; then 3000 will express the ratio of 1000 to 1, and this number 3000 will express the number of small ratios of the 1000th root of 10 to 1 contained in the ratio of 1000 to 1; and so on for any larger numbers, as 10,000, 100,000, or 10,000,000, &c. Thus, if instead of 1000 we make 10,000,000 the representative of the ratio of 10 to 1, then the unit will represent a very small ratio, of which there are 10,000,000 contained betwixt 1 and 10, and which ratio could not really be had without extracting a root which involved

in itself, 10,000,000 of times would only make up 10; which root may perhaps be most intelligibly expressed thus, $10,000,000\; \backslash sqrt\{10\}$. If the ratio of 10 to 1 contained 10,000,000 of these roots, it is evident that the ratio of 100 to 1 must contain 20,000,000, that of 1000 would have 30,000,000, &c.; of consequence, the ratio of 100 to 1 will be expressed by 20,000,000, of 1000 to 1 by 30,000,000, &c.—Hence, as these artificial numbers represent the ratios of natural numbers to unity, or are proportional to them, they are very properly called the logarithms of these numbers, or the numbers of their ratios; because they really do express this number of ratios.

The relation of logarithms to natural numbers may perhaps more intelligibly be explained by two series of numbers, one in an arithmetical, and the other in geometrical, progression. Thus,

In either of these series it is evident, that by adding any two terms of the upper line together, a number will be had which indicates that produced by multiplying the corresponding terms of the lower line. Thus, in the first two series, suppose we wish to know the product of $4\; \backslash times\; 32$. In the upper line we find 2 standing over the number 4, and 5 over 32; adding therefore 5 to 2 we find 7, the sum of this addition, standing over 128, the product of the two numbers. In like manner, if we wish to divide 256 by 8, from the number which stands over 256, viz. 8, subtract that which stands over 8, viz. 3; the remainder 5, which stands over 32, shows that the latter is the quotient of 256 divided by 8. Let it be required to involve 4 as high as the biquadrate or 4th power: Multiply 2, the number which stands over 4, by the index of the power to which the number is to be involved; which index is 4: the product 8, standing over 256, shows that this last number is the biquadrate of 4 required. Lastly, let it be required to extract the cube root of 64; divide the number 6, which stands over 64, by 3, the index of the root you wish to extract; the quotient 2, standing over 4, shows that 4 is the root sought.

These examples are sufficient to show the great utility of logarithms in the most tedious and difficult parts of arithmetic. But though it is thus easy to frame a table of logarithms for any series of numbers going on in geometrical progression, yet it must be far more difficult to frame a general table in which the logarithms of every possible series of geometricals shall correspond with each other. Thus, though in the above series we can easily find the logarithm of 4, 8, &c. we cannot find that of 3, 6, 9, &c.; and if we assume any random numbers for them, they will not correspond with those which have already been assumed for 4, 8, 16, &c. In the construction of every table, however, it was evident, that the arithmetical or logarithmic series ought to begin with 0; for if it began with unity, then the sum of the logarithms of any two numbers must be diminished by unity before we could find the logarithm of the product. Thus,

Here let it be required to multiply 4 by 16; the number 3 standing over 4, added to 5 which stands over 16, gives 8 which stands over 128: but this is not just; so that we must diminish the logarithm by 1, and then the number 7 standing over 64 shows the true product. In like manner it appears, that as we descend below unity in a logarithmic table, the logarithms themselves must begin in a negative series with respect to the former; and thus the logarithm of 0 will always be infinite; negative, if the logarithms increase with the natural numbers; but positive, if they decrease. For as the geometrical series must diminish by infinite divisions by the common ratio, the arithmetical one must decrease by infinite subtractions, or increase by infinite additions of the common difference.

This property of numbers was not unknown to the ancient mathematicians. It is mentioned in the works of Euclid; and Archimedes made great use of it in his Arenarius, or treatise on the number of the sands: and it is probable that logarithms would have been much sooner invented, had the real necessity for them been sooner felt; but this did not take place till the end of the 16th century, when the construction of trigonometrical tables, and solution of perplexed astronomical problems, rendered them absolutely indispensable.

About this time it is probable that many people wished to see such tables of numbers, and were making attempts to construct them; but the invention is certainly due to Lord Napier, baron of Merchiston in Scotland. The invention is by some indeed ascribed to Longomontanus; but with very little probability, as he never published any thing of the kind, nor laid claim to the invention, though he lived to see the publication of Baron Napier's tables. Concerning this invention we are told, that "one Dr Craig a Scotchman, coming out of Denmark into his own country, called upon Baron Napier, and told him of an invention of Longomontanus in Denmark, to save the trouble of the tedious multiplication and division in astronomical calculations; but could give no farther account of it than that it was by proportionable numbers. From this slight hint the baron immediately set about the work; and by the time that Dr Craig returned to call upon him, he had prepared a rude draught of it, which he called Canon mirabilis Logarithmorum; and this draught, with some alterations, was printed in 1614.

According to Kepler, one Juste Byrge, assistant astronomer to the landgrave of Hesse, either invented or projected logarithms long before Baron Napier, and composed a table of sines for every two seconds of the quadrant; though, by reason of his natural reservedness, he never published any thing to the world. But whatever might have been in this, it is certain that the world is indebted for logarithms to Baron Napier, who died in the year 1618. This nobleman likewise made considerable improvements in trigonometry; and the frequent numerical computations he had occasion for in this branch, undoubtedly contributed to his invention of the logarithms, that he might save part of the trouble in these calculations. His book published in 1614 was intitled Mirifici Logarithmorum Canonis descriptio. At this time he did not publish his method of constructing the numbers until

the sense of the learned should be known. In other respects the work is complete, containing all the logarithms of the natural numbers to the usual extent of logarithmic tables; with the logarithmic sines, tangents, and secants, for every minute of the quadrant, directions for using the tables, &c.

This work was published in Latin; but was afterwards translated into English by Mr Edward Wright, inventor of the principles of what has been falsely called Mercator's Sailing. The translation was sent to his lordship at Edinburgh, and returned with his approbation and some few additions. It was published in 1616, after Mr Wright's death, with a dedication to the East India Company, by his son Samuel Wright, and a preface by Mr Briggs, who afterwards distinguished himself so much in bringing logarithms to perfection. In this translation Mr Briggs also gave the description and draught of a scale invented by Mr Wright, as well as other methods invented by himself, for finding the intermediate proportional numbers; the logarithms already found having been only printed for such numbers as were the natural sines of each minute.

Mr Wright's translation was reprinted in 1618, with a new title-page, and the addition of 16 pages of new matter, "showing the method of calculating triangles, as well as a method of finding out such lines and logarithms as are not to be found in the canons."

Next year John Speidell published his New Logarithms, in which were some remedies for the inconveniences attending Lord Napier's method. The same year also Robert Napier, the Baron's son, published a new edition of his father's book, entitled Canonis Logarithmorum Descriptio; with another concerning the method of constructing them, which the Baron had promised; together with some other miscellaneous pieces, which his father had likewise composed along with Mr Briggs. In 1620 also, a copy of these works was printed at Lyons in one volume, by Bartholomew Vincent a bookseller there; but this publication seems to have been but little known, as Wingate, who carried logarithms to France four years after, is said to have been the first who introduced them into that country.

The Curfus Mathematicus published at Cologne in 1618 or 1619 by Benjamin Ursius, mathematician to the elector of Brandenburg, contains a copy of Napier's logarithms, together with some tables of proportional parts. In 1624 he published his Trigonometria, with a table of natural sines and their logarithms, according to Lord Napier's method, to every ten seconds in the quadrant. The same year a book on logarithms was published at Marburg by the celebrated Kepler, of the same kind with those of Napier. Both of these begin at 90° or the end of the quadrant; and, while the sines decrease, the logarithms gradually increase; till at the beginning of the quadrant, or 0, the logarithm is infinite. The only difference betwixt the logarithms of Napier and Kepler is, that in the former the arc is divided into equal parts, differing by one minute each; and consequently their sines to which the logarithms are adapted are intermediate numbers represented only by approximating decimals: but in Kepler's table, the radius is divided into equal parts; which are considered as perfect and terminate sines, having equal differences, and to which the logarithms are here adapted,

adapted. A treatise of some extent was prefixed to the work; in which the construction and use of logarithms is pretty largely treated of. In the year 1627 the same author introduced logarithms into his Rudolphine Tables, together with several others, viz. 1. A table similar to that already mentioned; only that the column of fines or absolute numbers is omitted, and another added in its stead, showing what part of the quadrant each arc is equal to; viz. the quotient arising from the division of the whole quadrant by each given arc, and expressed in integers and sexagesimal parts. 2. Napier's table of logarithmic fines to every minute of the quadrant; as also two other smaller tables adapted for the calculation of eclipses and the latitude of planets. In this work Justus Byrgius is mentioned as having invented logarithms before Napier.

The kind of logarithms now in use were invented by Mr Henry Briggs professor of geometry in Gresham college, London, at the time they were first discovered by Napier. As soon as the logarithms of Napier were published, Mr Briggs directed his attention to the study and improvement of them; and his employment in this way was announced in a letter to Mr Usher, afterwards the celebrated archbishop, in the year 1615. By him the scale was changed, and 0 was made the logarithm of 1; but lord Napier informed Mr Briggs that he had already thought of such a scheme, but chose rather to publish the logarithmic tables he had completed, and to let those alone until he should have more leisure as well as better health. At an interview betwixt Lord Napier and Mr Briggs, the present plan seems to have been settled; and in consequence of his lordship's advice, Mr Briggs made some alteration in the method of constructing his tables from that which he had begun. A correspondence also took place betwixt his lordship and Mr Briggs, which continued during the lifetime of the former. It appears, however, that, whether Mr Briggs thought of this alteration before lord Napier or not, he certainly was the person who first published it to the world; and some reflections have been thrown upon his lordship for not making any mention of the share which Mr Briggs had in it.

In 1617 Mr Briggs published his first thousand logarithms under the title of Logarithmorum Chilia Prima; and in 1620 Mr Edward Gunter published his Canon of Triangles, containing the artificial or logarithmic fines and tangents for every minute, to seven places of figures besides the index; the logarithm of the radius being 10.000, &c. These were the first tables of logarithmic fines, tangents, &c. which made their appearance upon the present plan; and in 1623 they were reprinted in his book de Sectors et Radiis, along with the Chilia Prima of Mr Briggs. The same year Mr Gunter applied these logarithms of numbers, fines, and tangents, to straight lines drawn on a ruler; and with these the proportions in common numbers, as well as in trigonometry, were solved by the mere application of a pair of compasses; a method founded upon this property, that the logarithms of the terms of equal ratios are equally different. The instrument is now well known by the name of the two-feet Gunter's Scale. By the same methods he also greatly improved the sector. He was also the first who used the word cotangent

for the sine of the complement of an arc; and he introduced the use of arithmetical complements into the logarithmical arithmetic. He is said also to have first suggested the idea of the logarithmic curve, so called because the segments of its axis are the logarithms of the corresponding ordinates.

The logarithmic lines were afterwards drawn in many other ways. Wingate, in 1627, drew them upon two separate rulers sliding by each other, in order to save the use of compasses in resolving proportions. In 1627 also, they were applied by Mr Oughtred to concentric circles; about 1650, in a spiral form, by one Mr Milburne of Yorkshire; and in 1657, they were applied on the present sliding-rule by Mr Seth Partridge.

The knowledge of logarithms was diffused in France by Mr Edmund Wingate, as already related, though not carried originally thither by him. Two small tracts were published by him in French, and afterwards an edition in English, all printed in London. In the first of these he mentions the use of Gunter's Ruler; and in the other that of Briggs's Logarithms, with the canon of artificial fines and tangents. There are likewise tables of these fines, tangents, and logarithms, copied from Gunter.

From the time that Mr Briggs first began to study the nature of logarithms, he applied to the construction of tables with such assiduity, that by the year 1624 he published his Arithmetica Logarithmica, containing the logarithms of 30,000 natural numbers to 14 places of figures besides the index; viz. from 1 to 20,000, and from 90,000 to 100,000, together with the differences of the logarithms. According to some, there was another Chiliad, viz. from 100,000 to 101,000; but this does not seem to be well authenticated. In the preface to this work, he gives an account of the alteration made in the scale by Lord Napier and himself; and earnestly solicits other persons to undertake the task of filling up the intermediate numbers, offering to give instructions, and to afford paper ready ruled for the purpose. He gives also instructions at large in the preface for the construction of logarithmic tables. Thus he hoped to get the logarithms of the other 70,000 natural numbers completed; while he himself, being now pretty far advanced in years, might be at liberty to apply to the canon of logarithmic fines, &c. which was as much wanted by mathematicians as the others. His wishes were accomplished by Adrian Vlacq or Flack of Gouda in Holland, who completed the numbers from 20 to 90,000; and thus the world was furnished with the logarithms of all natural numbers from 1 to 100,000; but those of Vlacq were only done to 10 places of figures. To these was added a table of artificial fines, tangents, and secants, to every minute of the quadrant. Besides the great work already mentioned, Mr Briggs completed a table of logarithmic fines and tangents for the 100th part of every degree, to 14 places of figures besides the index; and a table of natural fines for the same parts to 15 places, with the tangents and secants to 10 places, and the methods of constructing them. He designed also to have published a treatise concerning the uses and application of them, but died before this could be accomplished. On his death-bed he recommended this work to Henry Gellibrand professor of

astronomy in Gresham college, in which office he had succeeded Mr Gunter. Mr Briggs's tables above mentioned were printed at Gouda, and published in 1633; and the same year Mr Gellibrand added a preface with the application of logarithms to plane and spherical trigonometry, the whole being denominated Trigonometria Britannica: and besides the arcs in degrees and hundredth parts, has another table containing the minutes and seconds answering to the several hundredth parts in the first column.

The Trigonometria Artificialis of Vlacy contains the logarithmic fines and tangents to 10 places of figures, to which is added Briggs's first table of logarithms from 1 to 20,000, besides the index: The whole preceded by a description of the tables, and the application of them to plane and spherical trigonometry, chiefly extracted from Briggs's Trigonometria Britannica already mentioned. In 1635, Mr Gellibrand also published a work, intitled, An Institution Trigonometrical, containing the logarithms of the first 10,000 numbers, with the natural fines, tangents, and secants; and the logarithmic fines and tangents for degrees and minutes, all to seven places of figures besides the index; likewise other tables proper for navigation, with the uses of the whole. Mr Gellibrand died in 1636, in the 40th year of his age.

A number of other people have published books on logarithms, which we cannot now particularly enumerate. Some of the principal are:

1. A treatise concerning Briggs's logarithms of common numbers from 1 to 20,000, to 11 places of figures, with the logarithmic fines and tangents but only to eight places. By D. Henrion at Paris, 1626.

2. Briggs's logarithms, with their differences to 10 places of figures, besides the index for all numbers to 100,000; as also the logarithmic fines, tangents, and secants, for every minute of the quadrant, with the explanation and uses in English. By George Miller, Lond. 1631.

3. Trigonometria, by Richard Norwood, 1631; containing Briggs's logarithms from 1 to 10,000, as well as for the fines, tangents, and secants to every minute, both to several places of figures besides the index. The author complains very much of the unfair practices of both the former authors.

4. Directorium Generale Uranometricum; by Francis Bonaventure Cavalierus. Bologna, 1632. In this are Mr Briggs's tables of logarithmic fines, tangents, secants, and versed fines, each to eight places of figures for every second of the first 5 minutes, for every 5 seconds from 5 to 10 minutes, for every 20 seconds from 20 to 30 minutes, for every 30 seconds from 30 minutes to 1$\backslash frac\{1\}\{2\}$ degree, and for every minute in the rest of the quadrant. It contains also the logarithms of natural numbers from 1 to 10,000, with the first table of versed fines that ever was published. The author likewise gives the first intimation of the method of finding the arcs or spherical surface contained by various arcs described on the surface of a sphere.

5. In 1643 appeared the Trigonometria of the same author, containing the logarithms of the natural numbers from 1 to 1000, with their differences to eight places of figures; likewise a table of natural and logarithmic fines, tangents, and secants; the former to seven, the latter to eight, places of figures; viz. to

every 10^th of the first 30^th, to every 30^th from 30^th to 1^st, and the same for their complements, or backwards thro' the last degree of the quadrant; the intermediate 88^o being only to every minute.

6. Tabulae Logarithmice; by Mr Nathaniel Rowe, pastor of Benaire in Suffolk: Lond. 1633. In this work are contained Briggs's logarithms of natural numbers from 1 to 100,000, to eight places of figures; likewise the logarithmic fines and tangents to every 100th part of degrees to ten places.

7. Clavis Universalis Trigonometrica; Hamburg, 1634: containing tables of Briggs's logarithms from 1 to 2000; and of fines, tangents, and secants, for every minute, both for seven places.

8. Trigonometria Britannica, by John Newton, London, 1658. In this the logarithmic tables of natural numbers were reduced to their most convenient form; the author having availed himself of the labours of Wingate and Roe, uniting their several methods, and disposing of the whole as in the best logarithmic tables used at present. It contains likewise the logarithmic fines and tangents to eight figures besides the index; for every hundredth part of a degree, with the differences, and for thousandth parts in the first three degrees. He censures the unfair practices of some former publishers of logarithms; particularly of Vlacy already mentioned.

9. Mathesis Nova, by John Caramuel, 1670. This contained 1000 logarithms, both of the forms of Napier and Briggs, as well as 1000 of what he calls perfect logarithms, viz. those of Briggs's first method of construction; which differs from the last only in this, that the last increases, whilst the first decreases; the radix or logarithm of the ratio of 10 to 1 being the very same in both.

10. Sherwin's Mathematical Tables, published in 8vo, form the most complete collection of any; containing, besides the logarithms of all numbers to 101,000, the fines, tangents, secants, versed fines both natural and logarithmic, to every minute of the quadrant. The first edition was printed in 1706; but the third, published in 1742 and revised by Gardiner, is looked upon to be superior to any other. The fifth and last edition, published in 1771, is so incorrect, that no dependence can be placed upon it.

10. Tables of logarithms from 1 to 102,100, and for the fines and tangents to every 10 seconds of each degree in the quadrant; as also for the fines of the first 72 minutes to every single second, with other useful and necessary tables. By Gardiner, London, 1742. This work contains a table of logarithms, and three smaller tables to be used for finding the logarithms of numbers to 20 places of figures. Only a small number of these tables was printed, and that by subscription; and they are now in the highest esteem for accuracy and usefulness. An edition of these tables was printed at Avignon in France in 1770, with the addition of fines and tangents for every single second in the first four degrees, and a small table of hyperbolic logarithms, taken from a treatise upon fluxions by the late Mr Thomas Simson. The tables are to seven places of figures, but somewhat less correct than those published by Gardiner himself.

11. An Antilogarithmic Canon for readily finding the number corresponding to any logarithm, was begun by

the algebraist Mr Harriot, who died in 1621; and completed by Mr Walter Warner, the editor of Harriot's work, before 1640, but never was published for want of encouragement to print it. In 1714 a small specimen of such a canon appeared in the Philosophical Transactions for that year by Mr Long of Oxford; and in 1742 a complete Antilogarithmic Canon appeared by Mr James Dodson, in which the numbers corresponding to each logarithm from 1 to 100,000 are computed to 11 places of figures.

12. In 1783 were published M. Callet's tables at Paris; which for the elegance of the workmanship are much superior to any thing of the kind that ever appeared, though their accuracy is not esteemed equal to that of some others. The work is a neat volume small 8vo. It contains a treatise on logarithms, with their uses and application to various sciences; as trigonometry, astronomy, and navigation; a table of logarithms from 1 to 102,960, with the differences; a table of sines and tangents for every single second of the first two degrees, and for every 10 seconds of the rest of the quadrant; with tables of logarithical and hyperbolic logarithms, and some others for determining the longitude at sea.

SECT. II. Different methods of constructing Logarithms.

1. Napier's method.

THE logarithms first thought of by Lord Napier were not adapted to the natural series of arithmetical numbers 1, 2, 3, &c. because he did not then intend to adapt them to every kind of arithmetical calculation, but only to that particular operation which had called for their immediate construction, viz. the shortening of trigonometrical operations: he explained the generation of logarithms, therefore, in a geometrical way. Both logarithms, and the quantities to which they correspond, in his way, may be supposed to proceed from the motion of a point; which, if it moves over equal spaces in equal times, will produce a line increasing equally: but if, instead of moving over equal spaces in equal times, the point describes spaces proportional to its distances from a certain term, the line produced by it will then increase proportionally. Again, if the point moves over such spaces in equal times, as are always in the same constant ratio to the lines from which they are subtracted, or to the distance of that point at the beginning of the lines, from a given term in that line, the line so produced will decrease proportionally. Thus, let $ac$ be to $ao$, $ed$ to $eo$, $ef$ to $fo$, and $fg$ to $go$, always in a certain ratio, viz. that of $QR$ to $QS$, and let us suppose the point $P$ to set $f$ out from $a$, describing the distances $ac$, $ed$, $de$, &c. in equal spaces of time, then will the line $ao$ decrease proportionally.

In like manner, the line $oa$, (fig. 12.) increases proportionally, if the point $p$, in equal times, describes the spaces $ac$, $ed$, $de$, $fg$, &c. so that $ac$ is to $ao$, $ed$ to $eo$, $de$ to $do$, &c. in a constant ratio. If we now suppose a point $P$ describing the line $AG$ (fig. 4.) with an uniform motion, while the point $p$ describes a line increasing or decreasing proportionally, the line

$AP$, described by $P$, with this uniform motion, in the same time that $oa$, by increasing or decreasing proportionally, becomes equal to $op$, is the logarithm of $op$. Thus $AC$, $AD$, $AE$, &c. are the logarithms of $oc$, $od$, $oe$, &c. respectively: and $oa$ is the quantity whose logarithm is supposed equal to nothing.

We have here abstracted from numbers, that the doctrine may be the more general; but it is plain, that if $AC$, $AD$, $AE$, &c. be supposed 1, 2, 3, &c. in arithmetic progression; $oc$, $od$, $oe$, &c. will be in geometric progression; and that the logarithm of $oa$, which may be taken for unity, is nothing.

Lord Napier, in his first scheme of logarithms, supposes, that while $op$ increases or decreases proportionally, the uniform motion of the point $P$, by which the logarithm of $op$ is generated, is equal to the velocity of $p$ at $a$; that is, at the term of time when the logarithms begin to be generated. Hence logarithms, formed after this model, are called Napier's Logarithms, and sometimes Natural Logarithms.

When a ratio is given, the point $p$ describes the difference of the terms of the ratio at the same time. When a ratio is duplicate of another ratio, the point $p$ describes the difference of the terms in a double time. When a ratio is triplicate of another, it describes the difference of the terms in a triple time; and so on. Also, when a ratio is compounded of two or more ratios, the point $p$ describes the difference of the terms of that ratio in a time equal to the sum of the times in which it describes the differences of the terms of the simple ratios of which it is compounded. And what is here said of the times of the motion of $p$ when $op$ increases proportionally, is to be applied to the spaces described by $P$, in those times, with its uniform motion.

Hence the chief properties of logarithms are deduced. They are the measures of ratios. The excess of the logarithm of the antecedent above the logarithm of the consequent, measures the ratio of those terms. The measure of the ratio of a greater quantity to a lesser is positive; as this ratio, compounded with any other ratio, increases it. The ratio of equality, compounded with any other ratio, neither increases nor diminishes it; and its measure is nothing. The measure of the ratio of a lesser quantity to a greater is negative; as this ratio, compounded with any other ratio, diminishes it. The ratio of any quantity $A$ to unity, compounded with the ratio of unity to $A$, produces the ratio of $A$ to $A$, or the ratio of equality; and the measures of those two ratios destroy each other when added together; so that when the one is considered as positive, the other is to be considered as negative. By supposing the logarithms of quantities greater than $oa$ (which is supposed to represent unity) to be positive, and the logarithms of quantities less than it to be negative, the same rules serve for the operations by logarithms, whether the quantities be greater or less than $oa$. When $op$ increases proportionally, the motion of $p$ is perpetually accelerated; for the spaces $ac$, $ed$, $de$, &c. that are described by it in any equal times that continually succeed after each other, perpetually increase in the same proportion as the lines $oa$, $oc$, $od$, &c. When the point $p$ moves from $a$ towards $o$, and $op$ decreases proportionally, the motion of $p$ is perpetually retarded; for the spaces described by it in any equal times that

Construction of Logarithms continually succeed after each other, decrease in this case in the same proportion as $op$ decreases.

If the velocity of the point $p$ be always as the distance $op$, then will this line increase or decrease in the manner supposed by Lord Napier; and the velocity of the point $p$ being the fluxion of the line $op$, will always vary in the same ratio as this quantity itself. This, we presume, will give a clear idea of the genesis or nature of logarithms; but for more of this doctrine, see Maclaurin's Fluxions.

The construction of his tables of logarithms was first published in his posthumous work of 1619. The construction of his canon was chiefly effected by generating, in an easy manner, a series of proportional numbers, and their arithmeticals or logarithms; and then finding by proportion the logarithms of the natural fines from those of the nearest numbers among the original proportionals. Beginning then at the radius 10,000,000, he first constructs several descending geometrical series, of such a nature that they are quickly formed by an easy addition or subtraction, or division by 2, 10, 100, &c. His first table consists of proportionals in the ratio of 10,000,000 to 9,999,999; the method of doing which may be easily understood from the following example: Suppose it were required to find a series of descending proportionals in the ratio of 100 to 99; it may be done by adding two cyphers to each of the two first terms, and continually adding 1 to the decimal place farthest to the right hand. Thus the first term will be 100.00, the second 99.00, the third 98.01, the fourth 98.03, &c. Napier's first table contained 100 terms of a series, as we already mentioned, in the proportion of 10,000,000 to 9,999,999. The first term of which series was 10,000,000.000000; the second 9,999,999.000000; the third was 9,999,998.000001, and so on till the 100th term, which was 9,999,900.0004950. The second table consisted of 50 numbers nearly in the proportion of 100,000 to 99,999; and this was formed by substituting the units 1, 3, &c. in the third decimal place instead of the last place towards the right hand. The reason of constructing this table was, that he might have a series in the proportion of his first term of the former to the last term of it, viz. of 100,000 to 99,999; and the last of the second series was 9995001.222927. In all these series the method of finding the terms is exactly the same. Thus in the first example, where we begin with 100, each term decreases by the 100th part of the former; and this 100th part is found by removing the number two places of figures lower, and subtracting them from the former terms. Thus 99 is less than 100 by unity, which is the 100th part of the latter; the next term is less than 99 by the 100th part of 99, and is therefore 98.01. But the division by 100 can be performed without any trouble, only setting the decimal point two places farther forward, as that by 10 is performed by setting it one place farther forward; thus $9\; \backslash div\; 10\; =\; .9$; $99\; \backslash div\; 100\; =\; .99$. Now by subtracting 99 from 100, we have 98.01 for the third term of the series. To find the fourth term then, remove the decimal point two figures farther to the right hand, and subtract it from the former; and we have then 97.0299 for the fourth term of the series. Thus we see, that the number of decimal places must continually increase; but as in this series we want no more than two decimal

places instead of 97.0299, the term is made 97.03, as the nearest number which has only two decimal places, and differs from the truth only by one thousandth part. In like manner, in the long string of cyphers, the fourth term of the series differs somewhat, but very little, from the truth; and this must always be the case while the radius is supposed to consist of any finite number of parts; though, by going on for a very long time in this way, the error, by being continually repeated and augmented at every term, would at last become perceptible; and therefore none of these series are carried on to a very great length.

His next step was to construct a third table consisting of 69 columns, and each column of 21 numbers or terms in the continual proportion of 10000 to 9995; that is, nearly as the first term of the second table is to its last term. As this proportion is the 200th of the whole, the method of finding the terms will be by dividing each upper number by 2, and removing the figures of the quotient three places lower, and then subtracting them. In this way, however, it is proper to collect only the first column of 21 numbers, the last of which will be 9900473.5780; but the first, second, and third, &c. numbers in all the other columns are in the continual proportion of 100 to 99, or nearly of the first to the last in the first column; whence these are to be found by removing the figures two places lower, and then subtracting them, as has already been explained.

By these three tables, his lordship was furnished with about 1600 proportionals, nearly coinciding with all the natural series from 90 to 30 degrees. To obtain the logarithms of those proportionals, he demonstrated and applied some of the properties and relations of the numbers and logarithms; the principal of which are, 1. That the logarithm of any fine is greater than the difference between that fine and the radius, but less than that difference when increased in the proportion of the fine to the radius. 2. That the difference between the logarithms of two fines is less than the difference of the fines increased in the proportion of the lesser fine to the radius, but greater than the difference of the fines increased in the proportion of the greater fine to the radius. These properties now served him as theorems for finding the logarithms themselves in an easy manner. From the first of them it appeared, that the radius being 10,000,000, the first term of the table, the logarithm of 9,999,999, the second term, must be greater than the difference betwixt that term and the radius, which is 1, but less than the difference when increased in the proportion of the fine to the radius; but this proportion is only one ten millionth part, for $9,999,999\; \backslash times\; 1.0000001\; =\; 10,000,000$; whence the logarithm of the radius or 10,000,000 being 0, the logarithm of 9,999,999 the second term will be between 1 and 1.0000001, or very nearly 1.00000005, this being the arithmetical mean betwixt 1 and 1.0000001. This will also be the common difference betwixt every two succeeding terms in the first table; because all the terms there are in the same proportion of 10,000,000 to 9,999,999. Hence by the continual addition of this logarithm we have the logarithms of the whole series, and therefore that of the last term of the series viz. 9999900.0004950 will be 1.0000005.

The second table, as we have already said, consists

of a series of numbers in the continual proportion of 100000 to 99999 whence the first term being 10,000,000 the second will be 9,999,900; the difference betwixt this and the last term of the former series is .0004950. But by the second theorem, the difference between the logarithms of 9,999,900.0004950 and 9,999,900, the second term of the second table, will be less than .0004950, increased in the proportion of 99999 to 100000, but greater than .0004950, increased in the proportion of 9,999,900.0004950; that is to say, if we augment .0004950 by one hundred thousandth part, it will be greater than the difference betwixt the logarithms of the two terms. The limits, therefore, are here so extremely small, that we may account the difference betwixt the two terms and that of the logarithms themselves the same: adding therefore this difference .0004950 to 100.000005, we have 100.0005000 for the logarithm of the second term, and likewise for the common difference of all the logarithms of the terms of the second table. — Again, by the same theorem, the difference between the logarithms of this last proportional of the second table and the second term in the first column of the third table, will be found to be 1.2235287; which added to the last logarithm, gives 5001.2485387 for the logarithm of 9,995,000, the second term of the third table: and in a similar manner, by the same theorem, he finds the logarithms of all the other terms of the rest of the columns.

Thus our author completed what he calls his radical table, from which he found his logarithmic fines by taking, according to the second theorem, the sum and difference of each tabular fine, and the nearest number in the radical table. Annex then seven ciphers to the difference; divide the number by the sum, and half the quotient will be the difference between the logarithms of the tabular fine and radical number; and consequently, by adding or subtracting this difference to or from the logarithm of the natural number, we have the logarithmic fine required.

In this manner were completed the logarithmic fines from radius or fine of $90^\backslash circ$ to the half of it, or fine of $30^\backslash circ$. To complete the other $30^\backslash circ$, he observes, that the logarithm of the ratio of 2 to 1, or of one half the radius, is 6931469.22; that of the ratio of 4 to 1 is double of it; that of 8 to 1, triple of it, &c.; and thus going on to compute the logarithms of the ratio between 1 and 40, 80, 100, &c. to 10,000,000; then multiplying any given fine for an arc less than $30^\backslash circ$ by some of these numbers, he finds the product nearly equal to some number in the table; and then finds the logarithm by the second theorem as already directed.

Another, and much easier method, however, of performing the same thing is founded upon the following proportion, which he demonstrates, viz. that as half the radius is to the fine of half an arc, so is the cosine of the half arc to the fine of the whole arc; or as one half the radius is to the fine of any arc, so is the cosine of that arc to the fine of double the arc. Hence the logarithmic fine of an arc is found by adding the logarithms of half the radius and the fine of double the arc, and then subtracting the logarithmic cosine from the sum. In this way, he observes that the fines for full one half of the quadrant may be found, and the remainder by one easy division, or

addition and subtraction for each, as already directed.

§. 2. Kepler's method of construction.

THIS was founded upon principles nearly similar to that of Napier. He first of all erects a system of proportions, and the measures of proportion, founded upon principles purely mathematical; after which he applies these principles to the construction of his table, containing only the logarithms of 1000 numbers. The propositions on which his method is founded are in substance the following.

1. All equal proportions equal among themselves are expressed by the same quantity, be the terms many or few; as the proportion of 2, 4, 8, &c. in geometrical progression is expressed by 2; and of 2, 6, 18, 54, &c. by 3.

2. Hence the proportion of the extremes is composed of all the proportions of the intermediate terms; thus the proportion of 2 to 8 is compounded of that 2 to 4, and of 4 to 8.

3. The mean proportional betwixt two terms divides that proportion into two equal ones. Thus the proportion between 2 and 32 is divided by the mean proportional 8 into two equal proportions of 4; for 2 is to 8, as 8 is to 32.

4. In any number of proportionals regularly increasing, the means divide the proportion of the extremes into one more than their own number. Thus, in the series 2, 4, 8, 16, the proportion of the extremes 2 and 16 is by the two means 4 and 8, divided into three proportions, viz. that betwixt 2 and 4, 4 and 8, 8 and 16. In like manner, in the series 3, 6, 18, 54, 162, 486, the proportion betwixt 3 and 486 is divided by the four means into the five proportions of 3 to 6; 6 to 18; 18 to 54; 54 to 162; and 162 to 486.

5. The proportion betwixt any two terms is divisible into any number of parts, until these become less than any assignable quantity. Thus the proportion of 2 to 8 is divisible, by multiplying the two together and extracting the square root, into two parts by the number 4: by multiplying 2 and 4 together, and extracting the square root, and doing the same with 4 and 8, the proportion would be divided into four parts, viz. $2\; \backslash sqrt\{8\}\; \backslash cdot\; 4\; \backslash sqrt\{32\}\; \backslash cdot\; 8$; or in numbers, 2 : 2.813, &c. : 4 : 5.655, &c. : 8.

6. By dividing the ratios in this manner, the elementary part will become at last so small, that it may be denominated by the mere difference of terms of that element. This is evident from the diminution of the ratios or proportions already instanced: for the proportion between 2 and 2.813 is only 1.406, &c. and if we were to find a mean proportional betwixt 2 and 2.813, the ratio betwixt that proportional and 2 would be much less. But it must always be remembered, that such evanescent quantities, as they are called, cannot give us any conclusion with absolute exactness, however they may answer every useful purpose to us: for it is evident that neither mean proportional nor ratio can ever be found exactly; and therefore the error accumulated in all the operations must become very considerable, if any circumstance shall happen to make it appear.

7. In three continued proportionals, the difference

of the two first has to the difference between the two last the same proportion that the first term has to the second, or the second to the third. Thus, in the three terms, 4, 8, 16, the difference between the two first terms 4 and 8, viz. 4, is in proportion to 8; and the difference between the two last, as 4 is to 8, or 8 to 16.

8. In continued proportionals, the greatest terms have the greatest differences, and vice versa. Thus the difference between 8 and 16 is evidently greater than between 2 and 4 or 4 and 8.

9. If the difference betwixt the two greatest terms be made the measure of the proportion between them, the difference between any two others will be less than the true measure of their proportion. Thus in the series 4, 2, 1, $\backslash frac\{1\}\{2\}$, $\backslash frac\{1\}\{4\}$, &c. where the difference 2 betwixt the two greatest terms expresses their true proportion, it is plain, that the difference 1 betwixt 2 and 1 is less than their ratio, as well as between $\backslash frac\{1\}\{2\}$ and $\backslash frac\{1\}\{4\}$, &c.

10. In any series of proportionals, if the difference betwixt the greatest term and one not immediately next to it, be taken as the measure of the proportion, then the proportion betwixt the greatest term and any other greater than the term before taken, will be less than the difference of those terms; but the proportion which is between the greatest term and any one less than that first taken, will be greater than their difference. As proportionals of this kind do not readily occur, we shall, in order to avoid obscurity, show once for all, that there is a possibility of finding geometrical proportionals of such a nature, that the ratio may be equal to the difference betwixt the greatest and third, or any other term distant from it. Thus let us begin with any two numbers we please, suppose 9 and 10: though these are in the natural arithmetical proportion, yet if we make the ratio 1.111, they will also be geometrically proportional, and the series will run thus:

10 : 9 : 8.099 : 7.289 : 6.560 : 5.904, &c.

Here the difference betwixt the first and third terms is 1.901, which is greater than the ratio; that betwixt the second and fourth, viz. 1.711, is still greater, but nearer to it than the former; the difference between the third and fifth terms, viz. 1.539, still approximates, as does that between the fourth and sixth, viz. 1.385: and indeed by continuing this series only for two terms longer, the difference will become smaller than the ratio. It is not worth while, however, to seek for serieses of this kind, as the present proposition will now be sufficiently intelligible without any farther illustration.

11. If quantities be arranged according to the order of their magnitudes, and if any two successive proportions of these be equal, the three successive terms which constitute them will also be equal. Thus, if the two quantities 12 and 8 constitute the proportion $\backslash frac\{3\}\{2\}$, and each of them be lessened by 6, the half of 12, we have the proportion $\backslash frac\{3\}\{2\}$; which is more than double the original proportion; for $\backslash frac\{3\}\{2\}\; =\; 3$, and $\backslash frac\{1\}\{2\}\; \backslash times\; \backslash frac\{3\}\{2\}\; =\; \backslash frac\{3\}\{4\}\; =\; 2\backslash frac\{1\}\{2\}$.

12. When quantities are placed in the order of their magnitudes, if the intermediate magnitudes lying between any two terms be not among the mean propor-

nals which can be interposed betwixt these two terms, then these intermediates will not divide the proportion of those two terms into commensurable proportions. Thus in the magnitudes 343 : 216 : 125 : 64 : 27 : 8, neither of the two intermediate terms 125 and 64 are mean proportionals betwixt 27 and 216, nor do they divide the proportion betwixt these into commensurable parts.

13. All the proportions taken in order, which are between expressible terms that are in arithmetical proportion, are incommensurable to one another; as between 8, 13, and 18.

14. When quantities are placed in the order of their magnitude, if the difference between the two greatest be made the measure of their proportion, the difference between any two others will be less than the measure of their proportion; and if the difference between the two least terms be made the measure of their proportion, the differences of the rest will be greater than the measure of the proportion between their terms.

15. If the measure of proportion between the greatest exceed their difference, then the proportion of this measure to the difference will be less than that of a following measure to the difference of its terms; because proportionals have the same ratio.

16. If three equidifferent quantities are taken in order, the proportion between the extremes is more than double that betwixt the two greater terms. Hence it follows, that half the proportion of the extremes is greater than the proportion between the greatest terms, but less than the proportion of the two least.

17. If two quantities constitute a proportion, and each be lessened by half the greater, the remainder will constitute a proportion more than double the former.

18. If 1000 numbers follow one another in the natural order, 1000, 999, 998, &c. and by continual multiplication and extraction of the square root we find mean proportionals, and thus lisset, as it is called, the ratio between the two greatest, so that the parts into which the ratio is divided become ultimately smaller than the excess of proportion betwixt the next two over the former (for 998 bears a greater proportion to 999 than 999 bears to 1000); the measure of this very small part or element of the proportion may be supposed to be the difference between 1000 and that mean proportional which is the other term of the element. Thus, for the sake of an easy explanation, let us suppose the numbers to be 10, 9, 8, &c. the ratio of 9 to 10 is 1.11, that of 9 to 8 is 1.125, the difference between which is .014, which we may call the elementary part of the ratios. By six extractions of the square root we have the mean proportional 9.985, &c. differing from 10 by no more than .015, which is very near the element just mentioned. The number of parts into which the ratio is thus divided is expressed by the 6th power of 2 or 64. Dividing therefore the ratio between 9 and 10 or 1.11 by 64, we have .017 for the elementary part thus obtained; which near coincidence with the real element, and the difference between 10 and the mean proportional itself, shows that in large numbers we may take the difference between the mean proportional and greatest term for the elementary part without any sensible error.

Suppose now, that the proportion between 1000 and

and 998 be divided into twice the number of parts that the former was, it will be equally plain that the difference betwixt 1000 and the next mean proportional will be the measure of that element. Proceeding in like manner with the other numbers 1000 and 997, 1000 and 996, &c. it is evident, that by dividing into a proper number of parts, all the elements will be reduced to an equal degree of fineness, if we may so call it, and in calculations may be made use of without any fear of error.

19. The number of elementary parts being thus known which are contained in any proportion, it will be easy to find the ratios between those numbers which are in continued proportion to the first term of the series. Thus, having found the proportion between 1000 and 900,

we know also that of 1000 to 810, and 729; And from 1000 to 800, also 1000 to 640, and to 512; And from 1000 to 700, also 1000 to 490, and to 343; And from 1000 to 600, also 1000 to 360, and to 216; And from 1000 to 500, also 1000 to 250, and to 125.

Corol. Hence arises the precept for squaring, cubing, &c.; as also for extracting the square root, cube root, &c. out of the first figures of numbers. For it will be, As the greatest number of the chiliad as a denominator, is to the number proposed as a numerator, so is this to the square of the fraction, and so is this to the cube.

20. Prop. The proportion of a number to the first, or 1000, being known; if there be two other numbers in the same proportion to each other, then the proportion of one of these to 1000 being known, there will also be known the proportion of the other to the same 1000.

Corol. 1. Hence from the 15 proportions mentioned in prop. 18. will be known 120 others below 1000, to the same 1000.

For so many are the proportions, equal to some one or other of the said 15, that are among the other integer numbers which are less than 1000.

Corol. 2. Hence arises the method of treating the Rule-of-Three, when 1000 is one of the given terms.

For this is effected by adding to, or subtracting from, each other, the measures of the two proportions of 1000 to each of the other two given numbers, according as 1000 is, or is not, the first term in the Rule-of-three.

21. Prop. When four numbers are proportional, the

Let there be the fine 99970.1490 of an arc;
Its defect below radius is 29.8510 the covers. and less than logarithm fine;
Add the excess of the secant 29.8599

Sum 59.7109
its half or 29.8555 greater than the logarithm.

Therefore the logarithm is between { 29.8510 and
29.8555.

Precept 2. The logarithm of the fine being found, you will also find nearly the logarithm of the round or integer number which is next less than your fine with a fraction, by adding that fractional excess to the logarithm of the said fine.

Thus the logarithm of the fine 99970.149 is found to be about 29.854; if now the logarithm of the round N^o 184.

first to the second as the third to the fourth, and the proportions of 1000 to each of the three former are known, there will also be known the proportion of 1000 to the fourth number.

Corol. 1. By this means other chiliads are added to the former.

Corol. 2. Hence arises the method of performing the Rule-of-three, when 1000 is not one of the terms. Namely, from the sum of the measures of the proportions of 1000 to the second and third, take that of 1000 to the first, and the remainder is the measure of the proportion of 1000 to the fourth term.

Definition. The measure of the proportion between 1000 and any less number, as before described, and expressed by a number, is set opposite to that less number in the chiliad, and is called its logarithm, that is, the number (29.8510) indicating the proportion (29.8510) which 1000 bears to that number, to which the logarithm is annexed.

22. Prop. If the first or greatest number be made the radius of a circle, or finus totus; every less number considered as the cosine of some arc, has a logarithm greater than the versed sine of that arc, but less than the difference between the radius and secant of the arc; except only in the term next after the radius, or greatest term, the logarithm of which by the hypothesis is made equal to the versed sine.

That is, if CD be made the logarithm of AC, or the measure of the proportion of AC to AD; then the measure of the proportion of AB to AD, that is, the logarithm of AB, will be greater than BD, but less than EF. And this is the same as Napier's first rule in page 44.

23. Prop. The same things being supposed; the sum of the versed sine and excess of the secant over the radius, is greater than double the logarithm of the cosine of an arc.

Corol. The logarithm cosine is less than the arithmetic mean between the versed sine and the excess of the secant.

Precept 1. Any fine being found in the canon of fines, and its defect below radius to the excess of the secant above radius; then shall the logarithm of the fine be less than half that sum, but greater than the said defect or covered fine.

number 99970.000 be required, add 149 the fractional part of the fine to its logarithm, observing the point, thus,

29.854
149
the sum 30.003 } is the logarithm of the round number 99970.000 nearly.

24. Prop. Of three equidifferent quantities, the measure

Construction of Logarithms
sure of the proportion between the two greater terms, with the measure of the proportion between the two less terms, will constitute a proportion which will be greater than the proportion of the two greater terms, but less than the proportion of the two least.

Thus if AB, AC, AD, be three quantities having the equal differences BC, CD; and if the measure of the proportion of AD, AC be cd, and that of AC, AB, be be; then the proportion of cd to cb will be greater than the proportion of AC to AD, but less than the proportion of AB to AC.

25. Prop. The said proportion between the two measures is less than half the proportion between the extreme terms: that is, the proportion between bc, cd, is less than half the proportion between AB, AD.

Corol. Since therefore the arithmetical mean divides the proportion into unequal parts, of which the one is greater and the other less than half the whole; if it be enquired what proportion is between these proportions, the answer is, that it is a little less than the said half.

An example of finding nearly the limits, greater and less, to the measure of any proposed proportion.—It being known that the measure of the proportion between 1000 and 900 is 10536.05, required the measure of the proportion 900 to 800, where the terms 1000, 900, 800, have equal differences. Therefore as 9 to 10, so 10536.05 to 11706.72, which is less than 11778.30, the measure of the proportion 9 to 8. Again, as the mean proportional between 8 and 10 (which is 8.9442719) is to 10, so 10536.05 to 11779.66, which is greater than the measure of the proportion between 9 and 8.

Axiom. Every number denotes an expressible quantity.

26. Prop. If the 1000 numbers, differing by 1, follow one another in the natural order, and there be taken any two adjacent numbers, as the terms of some proportion; the measure of this proportion will be to the measure of the proportion between the two greatest terms of the chiliad, in a proportion greater than that which the greatest term 1000 bears to the greater of the two terms first taken, but less than the proportion of 1000 to the less of the said two selected terms.

So of the 1000 numbers taking any two successive terms, as 501 and 500, the logarithm of the former being 69114.92, and of the latter 69314.72, the difference of which is 199.80. Wherefore by the definition, the measure of the proportion between 501 and 500 is 199.80. In like manner, because the logarithm of the greatest term 1000 is 0, and of the next 999 is 100.05, the difference of these logarithms, and the measure of the proportion between 1000 and 999, is 100.05. Couple now the greatest term 1000 with each of the selected terms 501 and 500; couple also the measure 199.80 with the measure 100.05; so shall

VOL. X. Part I.

the proportion between 199.80 and 100.05 be greater than the proportion between 1000 and 501, but less than the proportion between 1000 and 500.

Corol. 1. Any number below the first 1000 being proposed, as also its logarithm; the differences of any logarithms antecedent to that proposed, towards the beginning of the chiliad, are to the first logarithm (viz. that which is assigned to 999) in a greater proportion than 1000 to the number proposed; but of those which follow towards the last logarithm, they are to the same in a less proportion.

Corol. 2. By this means the places of the chiliad may easily be filled up, which have not yet had logarithms adapted to them by the former propositions.

27. Prop. The difference of two logarithms, adapted to two adjacent numbers, is to the difference of these numbers in a proportion greater than 1000 bears to the greater of those numbers, but less than that of 1000 to the less of the two numbers.

This 27th proposition is the same as Napier's second rule.

28. Prop. Having given two adjacent numbers of the 1000 natural numbers, with their logarithmic indices, or the measures of the proportions which those absolute or round numbers constitute with 1000 the greatest; the increments or differences of these logarithms will be to the logarithm of the small element of the proportions, as the secants of the arcs whose cosines are the two absolute numbers is to the greatest number, or the radius of the circle: so that, however, of the said two secants, the less will have to the radius a less proportion than the proposed difference has to the first of all, but the greater will have a greater proportion, and so also will the mean proportional between the said secants have a greater proportion.

Thus if BC, CD be equal, also bd the logarithm of AB, and cd the logarithm of AC; then the proportion of be to cd will be greater than the proportion of AG to AD, but less than that of AF to AD, and also less than that of the mean proportional between AF and AG to AD.

Corol. 1. The same obtains also when the two terms differ, not only by the unit of the small element, but by another unit which may be ten fold, a hundred fold, or a thousand fold of that.

Corol. 2. Hence the differences will be obtained sufficiently exact, especially when the absolute numbers are pretty large, by taking the arithmetical mean between two small secants, or (if you will be at the labour) by taking the geometrical mean between two larger secants, and then by continually adding the differences, the logarithms will be produced.

Corol. 3. Precept. Divide the radius by each term of the assigned proportion, and the arithmetical mean (or still nearer the geometrical mean) between the quotients will be the required increment, which being added to the logarithm of the greater term, will give the logarithm of the less term.

Let there be given the logarithm of 700, viz. 35667.4948, to find the logarithm to 699.
Here radius divided by 700 gives 1428571, &c.
and divided by 699 gives 1430672, &c.
the arithmetic mean is 142.962
which added to 35667.4948

gives the logarithm to 699 35810.4568

Corol. 4. Precept for the logarithms of fines.

The increment between the logarithms of two fines is thus found: find the geometrical mean between the cosecants, and divide it by the difference of the fines, the quotient will be the difference of the logarithms.

EXAMPLE.

dis. 2909 geom. mean 2428 nearly.

The quotient 80000 exceeds the required increment of the logarithms, because the secants are here so large.

Appendix. Nearly in the same manner it may be shown, that the second differences are in the duplicate proportion of the first, and the third in the duplicate of the second. Thus, for instance, in the beginning of the logarithms, the first difference is 100.00000, viz. equal to the difference of the numbers 100000.00000 and 99900.00000; the second, or difference of the differences, 10000; the third 20. Again, after arriving at the number 50000.00000, the logarithms have for a difference 200.00000, which is to the first difference as the number 100000.00000 to 50000.00000; but the second difference is 40000, in which 10000 is contained four times; and the third 328, in which 20 is contained sixteen times. But since, in treating of new matters, we labour under the want of proper words, wherefore, lest we should become too obscure, the demonstration is omitted untried.

29. Prop. No number expresses exactly the measure of the proportion between two of the 1000 numbers constituted by the foregoing method.

30. Prop. If the measures of all proportions be expressed by numbers or logarithms; all proportions will not have assigned to them their due portion of measure, to the utmost accuracy.

31. Prop. If to the number 1000, the greatest of the chiliad, be referred others that are greater than it, and the logarithm of 1000 be made 0, the logarithms belonging to those greater numbers will be negative.

This concludes the first or scientific part of the work; the principles of which Kepler applies, in the second part, to the actual construction of the first 1000 logarithms, which is pretty minutely described. This part is intitled A very compendious method of constructing the Chiliad of Logarithms: and it is not improperly so called, the method being very concise and easy. The fundamental principles are briefly these: That at the beginning of the logarithms, their increments or differences are equal to those of the natural numbers: that the natural numbers may be considered as the decreasing cosines of increasing arcs: and that the secants of those arcs at the beginning have the same differences as the cosines, and therefore the same differences as the logarithms. Then, since the secants are the reciprocals of the cosines, by these principles and the third corol. to the twenty-eighth proposition, he establishes the fol-

lowing method of constituting the 100 first or smallest logarithms to the 100 largest numbers, 1000, 999, 998, 997, &c. to 900, viz. Divide the radius 1000, increased with seven ciphers, by each of these numbers separately, disposing the quotients in a table, and they will be the secants of those arcs which have the divisors for their cosines; continuing the division to the 8th figure, as it is in that place only that the arithmetical and geometrical means differ. Then, by adding successively the arithmetical means between every two successive secants, the sums will be the series of logarithms. Or, by adding continually every two secants, the successive sums will be the series of the double logarithms.

Besides these 100 logarithms thus constructed, he constitutes two others by continual bisection or extractions of the square root, after the manner described in the second postulate. And first he finds the logarithm which measures the proportion between 100000.00 and 97656.25, which latter term is the third proportional to 1024 and 1000, each with two ciphers; and this is effected by means of twenty-four continual extractions of the square root, determining the greatest term of each of twenty-four classes of mean proportionals; then the difference between the greatest of these means and the first or whole number 1000, with ciphers, being as often doubled, there arises 2371.6526 for the logarithm sought, which made negative is the logarithm of 1024. Secondly, the like process is repeated for the proportion between the numbers 1000 and 500, from which arises 69314.7193 for the logarithm of 500; which he also calls the logarithm of duplication, being the measure of the proportion of 2 to 1.

Then from the foregoing he derives all the other logarithms in the chiliad, beginning with those of the prime numbers 1, 2, 3, 5, 7, &c. in the first 100. And first, since 1024, 512, 256, 128, 64, 32, 16, 8, 4, 2, 1, are all in the continued proportion of 1000 to 500, therefore the proportion of 1024 to 1 is decuple of the proportion of 1000 to 500, and consequently the logarithm of 1 would be decuple of the logarithm of 500, if 0 were taken as the logarithm of 1024; but since the logarithm of 1024 is applied negatively, the logarithm of 1 must be diminished by as much: diminishing therefore 10 times the logarithm of 500, which is 693147.1928, by 2371.6526, the remainder 690775.5422 is the logarithm of 1, or of 100,000 what is set down in the table.

And because 1, 10, 100, 1000, are continued proportionals, therefore the proportion of 1000 to 1 is triple of the proportion of 1000 to 100, and consequently $\backslash frac\{1\}\{3\}$ of the logarithm of 1 is to be put for the logarithm of 100, viz. 230258.5141, and this is also the logarithm of decuplication, or of the proportion of 10 to 1. And hence

Construction of Logarithms
hence multiplying this logarithm of 100 successively by 2, 3, 4, 5, 6, and 7, there arise the logarithms to the numbers in the decuple proportion, as under.

Also if the logarithm of duplication, or of the proportion of 2 to 1, be taken from the logarithm of 1, there will remain the logarithm of 2; and from the logarithm of 2, taking the logarithm of 10, there remains the logarithm of the proportion of 5 to 1; which taken from the logarithm of 1, there remains the logarithm of 5. See the margin.

For the logarithms of other prime numbers, he has recourse to those of some of the first or greatest century of numbers, before found, viz. of 999, 998, 997, &c. And first, taking 960, whose logarithm is 4.082.2001; then by adding to this logarithm the logarithm of duplication, there will arise the several logarithms of all these numbers, which are in duplicate proportion continued from 960, namely 480, 240, 120, 60, 30, 15. Hence the logarithm of 30 taken from the logarithm of 10, leaves the logarithm of the proportion of 3 to 1; which taken from the logarithm of 1, leaves the logarithm of 3, viz. 5.8914.3106. And the double of this diminished by the logarithm of 1, gives 47.10, 53.0790 for the logarithm of 9.

Next, from the logarithm of 990, or $9\; \backslash times\; 10\; \backslash times\; 11$, which is 1005.0331, he finds the logarithm of 11; namely, subtract the sum of the logarithms of 9 and 10 from the sum of the logarithm of 990, and double the logarithm of 1, there remains 45.0986.0166 the logarithm of 11.

Again, from the logarithm of 980, or $2\; \backslash times\; 10\; \backslash times\; 7\; \backslash times\; 7$, which is 2020.2711, he finds 496184.5228 for the logarithm of 7.

And from 5129.3303 the logarithm of 950 or $5\; \backslash times\; 10\; \backslash times\; 19$, he finds 396331.6392 for the logarithm of 19.

In like manner the logarithm to 998 or $4\; \backslash times\; 13\; \backslash times\; 19$, gives the logarithm of 13; to 969 or $3\; \backslash times\; 17\; \backslash times\; 19$, gives the logarithm of 17; to 986 or $2\; \backslash times\; 17\; \backslash times\; 29$, gives the logarithm of 29; to 966 or $6\; \backslash times\; 7\; \backslash times\; 23$, gives the logarithm of 23; to 930 or $3\; \backslash times\; 10\; \backslash times\; 31$, gives the logarithm of 31.

And so on for all the primes below 100, and for many of the primes in the other centuries up to 900. After which he directs to find the logarithms of all numbers composed of these, by the proper addition and subtraction of their logarithms, namely, in finding the logarithm of the product of two numbers, from the sum of the logarithms of the two factors take the logarithm of 1, the remainder is the logarithm of the product. In this way he shows, that the logarithms of all numbers under 500 may be derived, except those of the following 36 numbers, namely 127, 149, 167, 173, 179, 211, 223, 251, 257, 263, 269, 271, 277, 281, 283, 293, 337, 347, 349, 353, 359, 367, 373, 379, 383, 389, 397, 401, 409, 419, 421, 431, 433, 439, 443, 449. Also, besides the composite numbers between 500 and 900, made up of the products of some numbers whose logarithms have been before determined, there will be 59 primes not composed of them; which with the 36 above mentioned make 95 num-

bers in all not composed of the products of any before them, and the logarithms of which he directs to be derived in this manner; namely, by considering the differences of the logarithms of the numbers interspersed among them; then by that method by which were constituted the differences of the logarithms of the smallest 100 numbers in a continued series, we are to proceed here in the discontinued series, that is, by prop. 28th, corol. 3d, and especially by the appendix to it, if it be rightly used, from whence those differences will be very easily supplied.

§ 3. Mr Briggs's Method.

THE methods principally made use of by this gentleman were published in Napier's posthumous work. Having supposed 0 to be the logarithm of 1, and 1 with any number of ciphers annexed, suppose 10 to be the logarithm of 10, this number is to be divided ten times by 5, which in a logarithmic number is equivalent to the extraction of the root of the fifth power; by which means he obtains the following numbers, viz. 2 with nine ciphers to it; 4 with eight ciphers; 8 with seven ciphers; 16 with six ciphers; 32 with five ciphers; 64 with four; 128000, 256000, 5120, and 1024. Dividing this last logarithm ten times by 2, we have a geometrical series of ten numbers; the first of which is 512, and the last 1. Thus 20 logarithms are obtained: but the labour of finding the numbers belonging to them is so excessive, that it is surprising how it could be undergone by any body. To obtain those corresponding to the first ten logarithms, the fifth root must be extracted ten times, and the square root as often, to obtain the numbers corresponding to the others. The power from which these extractions is made, must originally be 1, with a number of ciphers annexed. Other logarithms might be formed from these by adding them, and multiplying their corresponding numbers; but as this method, besides its excessive labour, would produce only an antilogarithmic canon like that of Mr Dodson already mentioned, other more easy and proper methods were thought of.

The next was by finding continually geometrical means, first between 10 and 1, and then between 10 and that mean, and so on, taking the arithmetical means between their corresponding logarithms. The operation is also facilitated by various properties of numbers and their logarithms, as that the products and quotients of numbers correspond to the sums and differences of their logarithms; that the powers and roots of numbers answer to the products and quotients of the logarithms by the index of the power or root. Thus having the logarithm of 2, we can have those of 4, 16, 256, &c. by multiplying the logarithms by 2, and squaring the numbers to as great an extent in that series as we please. If we have also that of 3, we can not only have those of 9, 81, 8561, &c. but of 6, 18, 27, and all possible products of the powers of 2 and 3 into one another, or into the numbers themselves. The following property may also be of use, viz. that if the logarithms of any two numbers are given, and each number be raised to the power denoted by the index of the other, the products will be equal. Thus,

Let the two numbers be 4 and 16; it is plain, that if

R 2 we

we raise 4 to the fourth power and 16 to the square, the products will be the same; for $16\; \backslash times\; 16\; =\; 256$, and $4\; \backslash times\; 4\; =\; 16$; $16\; \backslash times\; 4\; =\; 64$; and $64\; \backslash times\; 4\; =\; 256$.

Another method mentioned by Mr Briggs depends upon this property, that the logarithm of any number in this scale is 1 less than the number of places or figures contained in that power of that number whose exponent is the logarithm of 10, at least as to integral numbers; for Mr Briggs has shown that they really differ by a fraction. To this Mr Hutton adds the following; viz. that of any two numbers, as the greater is to the less, so is the velocity of the increment or decrement of the logarithms at the greater; "that is (says he), in our modern notation, as $X\; :\; Y\; :\; y\; :\; x$; where $x$ and $y$ are the fluxions of $X$ and $Y$.

In the treatise written upon the construction of logarithms by Mr Briggs himself, he observes, that they may be constructed chiefly by the two methods already mentioned, concerning which he premises several lemmata concerning the powers of numbers and their indices, and how many places of figures are in the products of numbers. He observes, that these products will consist of as many figures as there are in both factors, unless the first figures in each factor be expressed in one figure only, which sometimes happens, and then there will commonly be one figure less in the product than in the two factors. He observes also, that if in any series of geometricals, we take two terms, and raise one to the power denoted by the index of the other, or any number raised to the power denoted by the logarithm of the other, the product will be equal to this latter number raised to the power denominated by the logarithm of the former. Hence, if one of the numbers be 10, whose logarithm is 1 with any number of cyphers, then any number raised to the power whose index is the logarithm of that number, that is, the logarithm of any number in this scale where 1 is the logarithm of 10, is the index of that power of 10, which is equal to the given number. But the index of any integral power of 10 is one less than the number of places of figures it contains. Thus the square of 10, or 100, contains three places of figures, which is more by one, than 2 the index of the power; 1000, the cube of 10 contains four places, which is one more than the index, 3, of the power. Hence as the number of places of the powers of 10 are always exactly one more than the indices of those powers, it follows that the places of figures in the powers of any other number which is no integral power of 10, will not always be exactly one less in number than the indices of the powers. From these two properties is deduced the following rule for finding the logarithms of many prime numbers.

Find the 10th, 100th, 1000th, or any other power of a number, suppose 2, with the number of places of figures in it, then that number of figures shall always exceed the logarithm of 2, although the excess will be constantly less than 1; whence by proceeding to very high powers we will at last be able to obtain the logarithm of the number to great exactness.

Thus, the logarithm of 2, found by other methods, is known to be 30102999566389, &c. The tenth power of 2 is 1024; which containing four places of figures, gives 4 for the logarithm of 2, which exceeds it, though not quite by 1. The 20th power of 2, consisting of the 10th power multiplied into itself, by

its number of places ought to give the logarithm of 4; and according to the rule already laid down, should contain eight places of figures: but by reason of the cipher which stands in the second place, it is easy to see that it must contain only seven; which therefore gives seven for the logarithm of four. The logarithm of 16 is then expressed by the number of places of figures in the product of the 20th power of 2 into itself; and is therefore denominated by 13. That of 256 is denoted by the 80th power of 2, containing 25 places of figures. The logarithm of 2, therefore, having been already expressed by the 10th power of 2, will be again expressed by the 100th power. Adding, therefore, the number of places contained in the 80th power, viz. 25 to 7, the number of places contained in the 20th, we have 32 for the next expression of that logarithm. On account of the cipher which stands in the second place of one of the factors, however, we must deduct one from the number; and thus we have 31 for the logarithm of 2, which is a considerable approximation. Proceeding in this manner, at the 1000th power of 2, we have 302 for the logarithm of 2; at the 10,000th power we have 3011; at the 100,000th power, 30103; at the 1,000,000th, we have 301030; and at the 10,000,000th power, we obtain 3010300; which is as exact as is commonly expressed in the tables of logarithms; but by proceeding in the same manner we may have it to any degree of exactness we please. Thus, at the 100,000,000th power, we have 30103000; and at the 1,000,000,000th, the logarithm is 301029996, true to eight places of figures.

The only difficulty in this method is to find the number of places of figures in the different powers without multiplying them; but this may be determined by only multiplying the first five; or even the first three of the products will be sufficient to determine this; and the logarithms may thus be found with very great facility.

When the logarithms, however, are required to a very great degree of exactness, our author thinks that the method of mean proportionals is most eligible. This consists in finding a great number of mean proportionals betwixt 1 and the number proposed; that is, first extracting the square root of the number itself, then extracting the root of that root, &c. until the last root shall exceed 1 only by a very small decimal. Finding then the logarithm of this number by methods hereafter to be described, he multiplies it by the index of the power of 2, denoted by the number of extractions of the square root; and the result is the required logarithm of the given number. In this method, the number of decimal places contained in the last root ought to be double the number of true places required in the logarithm itself, and the first half of them ought to be cyphers; the integer being 1. To find out the first small number and its logarithm, our author begins with 10 and its logarithm 1; continually extracting the root of the former, and bisecting the latter, till he comes to the 54th root, and then finds, that at the 53d and 54th roots both natural numbers and logarithms bear the same proportion to each other, viz. that of 2 to 1. Thus,

Numbers.

53 | 1.00000.00000.00000.25563.82986.40064.70
54 | 1.00000.00000.00000.12781.91493.20031.35

L

Lo

5310.00000.00000.00000.1102.25024.62515.65404
5410.00000.00000.00000.0551.11512.31217.82702

If now by continual extraction and bisection we find any other small number, it will then be, as 12781, &c. is to 5551, &c. so is that other small decimal to the correspondent significant figures of its logarithm. To avoid, however, the excessive labour of such long multiplications and divisions, he reduces this ratio to another, the antecedent of which is 1. Thus, as 12781, &c. is to 5551, &c. so is 1 with as many ciphers annexed as precede the logarithms above mentioned, viz. 15, with another unit annexed to a 4th number, which will be the significant figures of the logarithm of the third term. The proportions then will be 12781.00.15551.00. = 1.00000.00000.00000.11434.9425190151864; this last number, with 17 ciphers prefixed, being the logarithm of the one immediately preceding it. Having therefore found by continual extraction any such small decimal as the above, multiply it by 4342, &c. and the product will be the corresponding logarithm of the last root.

Still, as the labour of so many extractions must be intolerably tedious, it became necessary to fall upon some contrivances to shorten such operations; and of these the following is an example.

Let the number of which we seek the logarithm be involved to such an height that the index of the power may be one, with either one or more ciphers next to it. Divide this power then by 1 with as many ciphers annexed as the power has significant figures after the first; or, supposing all the figures after the first to be decimals, the roots are extracted continually from this power, till the decimal becomes sufficiently small, as when the first 15 places are ciphers; then, multiplying the decimal by 43429, &c. we have the logarithm of this last root; which logarithm, multiplied by the like power of the number 2, gives the logarithm of the first number of which the extraction was begun. To this logarithm if we prefix 1, 2, 3, &c. according as this number was found by dividing the power by 10, 100, 1000, &c. and lastly, dividing the result by the index of that power, the quotient will be the required logarithm of the given prime number.

Thus to find by this method the logarithm of 2. Raise it first to the 10th power, which is 1024; then cutting off for decimals the last three figures, we continually extract the square root from 1,024 forty-seven times, which gives

1.00000.00000.00000.16851.60570.53949.77; the decimal part of which multiplied by 43429, &c. gives 0.00000.00000.00000.07318.55936.90623.9368 for its logarithm, which being continually doubled 47 times, or multiplied at once by the 47th power of 2, viz. 140737488355328, gives for the logarithm of the number 1024.01029.99566.39811.95265.27744, true to 17 or 18 places of decimals; then prefixing to this

number 3, because the division was made by 1000 (for cutting off the three places of decimals is the same as dividing by 1000), we have for the logarithm of 1024, 3.010299566, &c. as above. Lastly, dividing by 10, because 1024 is the 10th power of 2, we have the logarithm of 2 itself; viz. 0.30102, &c.

The involving of any number to a very high power is by no means a matter of such difficulty as might at first sight be imagined. A number multiplied by itself produces the square; the square multiplied by itself produces the biquadrate; the biquadrate multiplied by itself gives the eighth power, and the eighth power multiplied by the square produces the tenth. The tenth power multiplied by itself gives the 20th, and the 20th multiplied by itself the 40th. The eighth power divided by the original number gives the seventh; and the 40th power multiplied by the seventh gives the 47th power required.

The differential method of constructing logarithms was likewise invented by our author, and greatly shortens the labour of finding the mean proportionals. Mr Briggs, in the course of his calculations, had observed, that these proportionals, found by continual extraction of roots, gradually approach nearer and nearer to the halves of the preceding root; and that as many significant figures as there are ciphers before them, agree exactly in this proportion. Subtracting therefore each of these decimal parts, which he called A, or the first differences, from half the next preceding one, and by comparing together the remainders or second differences, called B, he found that the succeeding were always nearly equal to $\backslash frac\{1\}\{2\}$ of the next preceding ones; then taking the difference between each second difference and $\backslash frac\{1\}\{2\}$ of the preceding one, he found that these third differences, called C, were nearly in the continual ratio of 8 to 1; again taking the difference between each C and $\backslash frac\{1\}\{2\}$ of the next preceding, he found that these fourth differences, called D, were nearly in the continual ratio of 16 to 1; and so on, the 5th (E), 6th (F), &c. differences, being nearly in the continual ratio of 32 to 1, of 64 to 1, &c.: these plain observations being made, they very naturally and clearly suggested to him the notion and method of constructing all the remaining numbers from the differences of a few of the first, found by extracting the roots in the usual way. This will evidently appear from the annexed specimen of a few of the first numbers in the last example for finding the logarithm of 6; where after the 9th number the rest are supposed to be constructed from the preceding differences of each, as here shown in the 10th and 11th. And it is evident that, in proceeding, the trouble will become always less and less; the differences gradually vanishing, till at last only the first differences remain. And that generally each less difference is shorter than the next greater, by as many places as there are ciphers at the beginning of the decimal in the number to be generated from the differences.

He then concludes this chapter with an ingenious, but not obvious, method of finding the differences B, C, D, E, &c. belonging to any number, as suppose the 9th, from that number itself, independent of any of the preceding 8th, 7th, 6th, 5th, &c.;

Thus in the 9th number of the foregoing example, omitting the ciphers at the beginning of the decimals, we have

Consequently

and it is this: Raise the decimal A to the 2d, 3d, 4th, 5th, &c. powers; then will the 2d (B), 3d (C), 4th (D), &c. differences be as here below, viz.

which agree with the like differences in the foregoing specimen.

§ 4. Of Curves related to Logarithms.

SEVERAL other ingenious methods and improvements are laid down by our author in his treatise upon this subject; but as all these were attended with great labour, mathematicians still continued their efforts to facilitate the work; and it was soon perceived that some curves had properties analogous to logarithms. Edmund Gunter, it has been said, first gave the idea of a curve, whose abscissæ are in arithmetical progression, while the corresponding ordinates are in geometrical progression, or whose abscissæ are the logarithms of their ordinates; but it is not noticed in any part of his writings. The same curve was afterwards considered

considered by others, and named the logarithmic or logistic curve by Huygens in his Dissertatio de Causa Gravitationis, where he enumerates all the principal properties of this curve, showing its analogy to logarithms. Many other learned men have also treated of its properties; particularly Le Seur and Jacquier in their comment on Newton's Principia; Dr John Kiell in the elegant little tract on logarithms subjoined to his edition of Euclid's Elements; and Francis Maferes, Esq; curator baron of the exchequer, in his ingenious treatise on Trigonometry; in which books the doctrine of logarithms is copiously and learnedly treated, and their analogy to the logarithmic curve, &c. fully displayed. It is indeed rather extraordinary that this curve was not sooner announced to the public; since it results immediately from Baron Napier's manner of conceiving the generation of logarithms, by only supposing the lines which represent the natural numbers to be placed at right angles to that upon which the logarithms are taken. This curve greatly facilitates the conception of logarithms to the imagination, and affords an almost intuitive proof of the very important property of their fluxions, or very small increments, viz. that the fluxion of the number is to the fluxion of the logarithm, as the number is to the subtangent; as also of this property, that, if three numbers be taken very nearly equal, so that their ratios to each other may differ but a little from a ratio of equality; as for example, the 3 numbers 10,000,000, 10,000,001, 10,000,002, their differences will be very nearly proportional to the logarithms of the ratios of those numbers to each other: all which follows from the logarithmic arcs being very little different from their chords, when they are taken very small. And the constant subtangent of this curve is what was afterwards by Cotes called the modulus of the system of logarithms: and since, by the former of the two properties above mentioned, this subtangent is a fourth proportional to the fluxion of the number, the fluxion of the logarithm, and the number, this property afforded occasion to Mr Baron Maferes to give the following definition of the modulus, which is the same in effect as Cotes's, but more clearly expressed; namely, that it is the limit of the magnitude of a fourth proportional to these three quantities, viz. the difference of any two natural numbers that are very nearly equal to each other, either of the said numbers and the logarithm or measure of the ratio they have to each other. Or we may define the modulus to be the natural number at that part of the system of logarithms, where the fluxion of the number is equal to the fluxion of the logarithm, or where the numbers and logarithms have equal differences. And hence it follows, that the logarithms of equal numbers or of equal ratios, in different systems, are to one another as the moduli of those systems. Moreover, the ratio whose measure or logarithm is equal to the modulus, and thence by Cotes called the ratio modularis, is by calculation found to be the ratio of 2.718281828459, &c. to 1, or of 1 to .367879441171, &c.: the calculation of which number may be seen at full length in Mr Baron Maferes's treatise on the Principles of Life-annuities, p. 274 and 275.

The hyperbolic curve also afforded another source for developing and illustrating the properties and construction of logarithms. For the hyperbolic areas ly-

ing between the curve and one asymptote, when they are bounded by ordinates parallel to the other asymptote, are analogous to the logarithms of their abscissæ or parts of the asymptote. And so also are the hyperbolic sectors; any sector bounded by an arc of the hyperbola and two radii being equal to the quadrilateral space bounded by the same arc, the two ordinates to either asymptote from the extremities of the arc and the part of the asymptote intercepted between them. And although Napier's logarithms are commonly said to be the same as hyperbolic logarithms, it is not to be understood that hyperbolas exhibit Napier's logarithms only, but indeed all other possible systems of logarithms whatever. For, like as the right-angled hyperbola, the side of whose square inscribed at the vertex is 1, gives us Napier's logarithms; so any other system of logarithms is expressed by the hyperbola whose asymptotes form a certain oblique angle, the side of the rhombus inscribed at the vertex of the hyperbola in this case also being still 1, the same as the side of the square in the right-angled hyperbola. But the areas of the square and rhombus, and consequently the logarithms of any one and the same number or ratio, will differ according to the sine of the angle of the asymptotes. And the area of the square or rhombus, or any inscribed parallelogram, is also the same thing as what was by Cotes called the modulus of the system of logarithms; which modulus will therefore be expressed by the numerical measure of the sine of the angle formed by the asymptotes, to the radius 1; as that is the same with the number expressing the area of the said square or rhombus, the side being 1: which is another definition of the modulus to be added to those we before remarked above in treating of the logarithmic curve. And the evident reason of this is, that in the beginning of the generation of these areas from the vertex of the hyperbola, the nascent increment of the abscissa drawn into the altitude 1, is to the increment of the area, as radius is to the sine of the angle of the ordinate and abscissa, or of the asymptotes; and at the beginning of the logarithms, the nascent increment of the natural numbers is to the increment of the logarithms as 1 is to the modulus of the system. Hence we easily discover, that the angle formed by the asymptotes of the hyperbola, exhibiting Briggs's System of Logarithms, will be $25^\{\backslash circ\}\; 44\text{'}\; 25\backslash frac\{1\}\{2\}\text{'}\text{'}$; this being the angle whose sine is 0.4342944819, &c. the modulus of this system.

Or indeed any one hyperbola, as has been remarked by Mr Baron Maferes, will express all possible systems of logarithms whatever; namely, if the square or rhombus inscribed at the vertex, or, which is the same thing, any parallelogram inscribed between the asymptotes and the curve at any other point, be expounded by the modulus of the system; or, which is the same, by expounding the area, intercepted between two ordinates which are to each other in the ratio of 10 to 1, by the logarithm of that ratio in the proposed system.

As to the first remarks on the analogy between logarithms and the hyperbolic spaces; it having been shown by Gregory St Vincent, in his Quadratura Circuli et Sectionum Coni, published at Antwerp in 1647, that if one asymptote be divided into parts in geometrical progression, and from the points of division or-

ordinates.

dinate be drawn parallel to the other asymptote, they will divide the space between the asymptote and curve into equal portions; from hence it was shown by Mercennus, that, by taking the continual sums of those parts, there would be obtained areas in arithmetical progression, adapted to abscisses in geometrical progression, and which therefore were analogous to a system of logarithms. And the same analogy was remarked and illustrated soon after by Huygens and many others, who show how to square the hyperbolic spaces by means of the logarithms. There are likewise many other geometrical figures which have properties analogous to logarithms; such as the equiangular spiral, the figures of the tangents and secants, &c.

§ 5. Mercator's Method.

THIS is purely arithmetical, and is founded upon the idea of logarithms already mentioned; viz. that they are the measures of ratios, and express the number of ratuncule contained in any ratio into which it may be divided. Having shown then that these logarithms, or numbers of small ratios, or measures of ratios, may be all properly represented by numbers; and that of 1, or the ratio of equality, the logarithm or measure being always 0, the logarithm of 10, or the measure of the ratio of 10 to 1, is most conveniently represented by 1 with any number of ciphers; he then proceeds to show how the measures of all other ratios may be found from this last supposition. And he explains the principles by the two following examples.

First, to find the logarithm of 100.5, or to find how many ratuncule are contained in the ratio of 100.5 to 1, the number of ratuncule in the decuple ratio, or ratio of 10 to 1, being 10,000,000.

The given ratio 100.5 to 1 he first divides into its parts; namely, 100.5 to 100, 100 to 10, and 10 to 1; the last two of which being decuples, it follows that the characteristic will be 2, and it only remains to find how many parts of the next decuple belong to the first ratio of 100.5 to 100. Now if each term of this ratio be multiplied by itself, the products will be in the duplicate ratio of the first terms, or this last ratio will contain a double number of parts; and if these be multiplied by the first terms again, the ratio of the last products will contain three times the number of parts, and so on; the number of times of the first parts contained in the ratio of any like powers of the first terms, being always denoted by the exponent of the power. If therefore the first terms 100.5 and 100 be continually multiplied till the same powers of them have to each other a ratio whose measure is known; as suppose the decuple ratio 10 to 1, whose measure is 10,000,000: then the exponent of that power shows what multiple this measure 10,000,000 of the decuple ratio is of the required measure of the first ratio 100.5 to 100; and consequently dividing 10,000,000 by that exponent, the quotient is the measure of the ratio 100.5 to 100 sought. The operation for finding this he sets down as here follows; where the several multiplications are all performed in the contracted way by inverting the figures of the multiplier, and retaining only the first number of decimals in each product.

N^o 184.

This power being greater than the decuple of the like power of 100, which must always be 1 with ciphers, refuse therefore the 256th power, and multiply it not by itself but by the next before it, viz. by the 128th, thus,

This power again exceeding the same power of 100 more than 10 times, he therefore draws the same 448th not into the 32d but the next preceding, thus,

This being again too much, instead of the 16th draw it into the 8th or next preceding, thus,

Which

Since therefore the 462d power of 100.5 is greater, and the 461st power is less, than the decuple of the same power of 100; he finds that the ratio of 100.5 to 100 is contained in the decuple more than 461 times, but less than 462 times. Again,

Since $\backslash left\backslash \{\; \backslash begin\{array\}\{l\}\; 460\; \backslash \backslash \; 461\; \backslash \backslash \; 462\; \backslash end\{array\}\; \backslash right\backslash \}$ power $\backslash left\backslash \{\; \backslash begin\{array\}\{l\}\; 9916193\; \backslash \backslash \; 9965774\; \backslash \backslash \; 10015603\; \backslash end\{array\}\; \backslash right\backslash \}$ and the differences $\backslash left\backslash \{\; \backslash begin\{array\}\{l\}\; 49581\; \backslash \backslash \; 49829\; \backslash end\{array\}\; \backslash right\backslash \}$ nearly equal;

therefore the proportional part which the exact power, or 1000000, exceeds the next less 9965774, will be easily and accurately found by the Golden Rule, thus:

The just power - - - 1000000
and the next less - - - 9965774
the difference - - - 34226; then,

As 49829 the dif. between the next less and greater,
: To 34226 the dif. between the next less and just,
:: So is 10000: to 6868, the decimal parts; and therefore the ratio of 100.5 to 100, is 461.6868 times contained in the decuple or ratio of 10 to 1. Dividing now 1,000,000, the measure of the decuple ratio, by 461.6868, the quotient 2,00216597 is the measure of the ratio of 100.5 to 100; which being added to 2, the measure of 100 to 1, the sum 2,00216597 is the measure of the ratio of 100.5 to 1, that is, the log. of 100.5 is 2,00216597.

In the same manner he next investigates the log. of 99.5, and finds it to be 1,99782307.

A few observations are then added, calculated to generalize the consideration of ratios, their magnitude and affections. It is here remarked, that he considers the magnitude of the ratio between two quantities as the same, whether the antecedent be the greater or the less of the two terms; so the magnitude of the ratio of 8 to 5 is the same as of 5 to 8; that is, by the magnitude of the ratio of either to the other is meant the number of ratimunculae between them, which will evidently be the same whether the greater or less term be the antecedent. And he farther remarks, that of different ratios, when we divide the greater term of each ratio by the less, that ratio is of the greater mass or magnitude which produces the greater quotient, et vice versa; although those quotients are not proportional to the masses or magnitudes of the ratios. But when he considers the ratio of a greater term to a less, or of a less to a greater, that is to say, the ratio of greater or less inequality, as abstracted from the magnitude of the ratio, he distinguishes it by the word affection, as much as to say greater or less affection, something in the manner of positive and negative quantities, or such as are affected with the signs + and -. . . . . The remainder of this work he delivers in several propositions, as follows:

Prop. 1. In subtraction from each other two quantities of the same affection, to wit, both positive, or both negative; if the remainder be of the same affection with the two given, then is the quantity subtracted the less of the two, or expressed by the less number; but if the contrary, it is the greater.

Prop. 2. In any continued ratios, as $\backslash frac\{a\}\{a+b\}$, $\backslash frac\{a+b\}\{a+2b\}$

$\backslash frac\{a+2b\}\{a+3b\}$, &c. (by which is meant the ratios of $a$ to $a+b$, $a+b$ to $a+2b$, $a+2b$ to $a+3b$, &c.) of equidifferent terms, the antecedent of each ratio being equal to the consequent of the next preceding one, and proceeding from less terms to greater; the measure of each ratio will be expressed by a greater quantity than that of the next following; and the same through all their orders of differences, namely, the 1st, 2d, 3d, &c. differences; but the contrary, when the terms of the ratios decrease from greater to less.

Prop. 3. In any continued ratios of equidifferent terms, if the 1st or least be $a$, the difference between the 1st and 2d $b$, and $c$, $d$, $e$, &c. the respective first term of their 2d, 3d, 4th, &c. differences; then shall the several quantities themselves be as in the annexed scheme; where each term is composed of the first term together with as many of the differences as it is distant from the first term, and to those differences joining, for coefficients, the numbers in the sloping or oblique lines contained in the annexed table of figurate numbers; in the same manner, he observes, as the same figurate numbers complete the powers raised from a binomial root, as had long before been taught by others. He also remarks, that this rule not only gives any one term, but also the sum of any number of successive terms from the beginning, making the 2d coefficient the 1st, the 3d the 2d, and so on; thus, the sum of the first 5 terms is $5a\; +\; 10b\; +\; 10c\; +\; 5d\; +\; e$.

1st term - - - $a$
2d - - - $a\; +\; b$
3d - - - $a\; +\; 2b\; +\; c$
4th - - - $a\; +\; 3b\; +\; 3c\; +\; d$
5th - - - $a\; +\; 4b\; +\; 6c\; +\; 4d\; +\; e$
&c. &c.

In the 4th prop. it is shown, that if the terms decrease, proceeding from the greater to the less, the same theorems hold good, by only changing the sign of every other term, as below.

1st term - - - $a$
2d - - - $a\; -\; b$
3d - - - $a\; -\; 2b\; +\; c$
4th - - - $a\; -\; 3b\; +\; 3c\; -\; d$
5th - - - $a\; -\; 4b\; +\; 6c\; -\; 4d\; +\; e$
&c. &c.

Prop. 6th and 7th, treat of the approximate multiplication and division of ratios, or, which is the same thing, the finding nearly any powers or any roots of a given fraction, in an easy manner. The theorem for raising

Constructing any power, when reduced to a simpler form, is raising any power, when reduced to a simpler form, is this, the $m$ power of $\backslash frac\{a\}\{b\}$, or $\backslash left(\backslash frac\{a\}\{b\}\backslash right)^m$ is $\backslash frac\{s\; \backslash pm\; md\}\{s\; \backslash mp\; md\}$ nearly, where $s$ is $a\; +\; b$, and $d\; =\; a\; -\; b$, the sum and difference of the two numbers, and the upper or under signs take place according as $\backslash frac\{a\}\{b\}$ is a proper or an improper fraction, that is, according as $a$ is less or greater than $b$. And the theorem for extracting the $m$th root of $\backslash frac\{a\}\{b\}$ is $\backslash sqrt[m]\{\backslash frac\{a\}\{b\}\}$ or $\backslash left(\backslash frac\{a\}\{b\}\backslash right)^\{\backslash frac\{1\}\{m\}\}\; =\; \backslash frac\{ms\; \backslash mp\; d\}\{ms\; \backslash pm\; d\}$ nearly; which latter rule is also the same as the former, as will be evident by substituting $\backslash frac\{1\}\{m\}$ instead of $m$ in the first theorem. So that universally $\backslash left(\backslash frac\{a\}\{b\}\backslash right)^\{\backslash frac\{1\}\{n\}\}$ is $\backslash frac\{ns\; \backslash mp\; md\}\{ns\; \backslash pm\; md\}$ nearly. These theorems, however, are nearly true only in some certain cases, namely, when $\backslash frac\{a\}\{b\}$ and $\backslash frac\{m\}\{n\}$ do not differ greatly from unity. And in the 7th prop. the author shows how to find nearly the error of the theorems.

In the 8th prop. it is shown, that the measures of ratios of equidifferent terms, are nearly reciprocally as the arithmetical means between the terms of each ratio. So of the ratios $\backslash frac\{1\}\{2\},\; \backslash frac\{1\}\{3\},\; \backslash frac\{1\}\{4\}$, the mean between the terms of the first ratio is $\backslash frac\{3\}\{4\}$, of the 2d $\backslash frac\{4\}\{3\}$, of the 3d $\backslash frac\{5\}\{2\}$, and the measures of the ratios are nearly as $\backslash frac\{1\}\{2\},\; \backslash frac\{1\}\{3\},\; \backslash frac\{1\}\{4\}$.

From this property he proceeds, in the 9th prop. to find the measure of any ratio less than $\backslash frac\{1\}\{100\}$, which has an equal difference (1) of terms. In the two examples mentioned near the beginning, our author found the logarithm or measure of the ratio, of $\backslash frac\{1\}\{100\}$, to be 21769, and that of $\backslash frac\{1\}\{1000\}$, to be 21659; therefore the sum 43429 is the logarithm of $\backslash frac\{1\}\{100\}$, or $\backslash frac\{1\}\{100\}\; \backslash times\; \backslash frac\{1\}\{1000\}$; or the logarithm of $\backslash frac\{1\}\{100\}$ is nearer 43430, as found by other more accurate computations. Now, to find the logarithm of $\backslash frac\{1\}\{100\}$, having the same difference of terms (1) with the former; it will be, by prop. 8. as 100.5 (the mean between 101 and 100) : 100 (the mean between 99.5 and 100.5) :: 43430 : 43213 the logarithm of $\backslash frac\{1\}\{100\}$, or the difference between the logarithms of 100 and 101. But the logarithm of 100 is 2; therefore the logarithm of 101 is 2.0043213. Again, to find the logarithm of 102, we must first find the logarithm of $\backslash frac\{101\}\{102\}$; the mean between its terms being 101.5, therefore as 101.5 : 100 :: 43430 : 42788 the logarithm of $\backslash frac\{101\}\{102\}$, or the difference of the logarithms of 101 and 102. But the logarithm of 101 was found above to be 2.0043213; therefore the logarithm of 102 is 2.0086001. So that dividing continually 868596 (the double of 434298 the logarithm of $\backslash frac\{1\}\{100\}$ or $\backslash frac\{1\}\{1000\}$) by each number of the series 201, 203, 205, 207, &c. then add 2

to the 1st quotient, to the sum add the 2d quotient, and so on, adding always the next quotient to the last sum, the several sums will be the respective logarithms of the numbers in this series, 101, 102, 103, 104, &c.

The next, or prop. 10th, shows that, of two pair of continued ratios, whose terms have equal differences, the difference of the measures of the first two ratios is to the difference of the measures of the other two, as the square of the common term in the two latter is to that in the former, nearly. Thus, in the four ratios $\backslash frac\{a\}\{a+b\},\; \backslash frac\{a+b\}\{a+2b\},\; \backslash frac\{a+3b\}\{a+4b\},\; \backslash frac\{a+4b\}\{a+5b\}$, as the measure of $\backslash frac\{aa+2ab\}\{a+b^2\}$ (the difference of the first two, or the quotient of the two fractions): the measure of $\backslash frac\{aa+8ab+15bb\}\{a+4b^2\}$ is $\backslash sqrt\{a+4b^2\}\; :\; \backslash sqrt\{a+b^2\}$, nearly.

In prop. 11. the author shows that similar properties take place among two sets of ratios, consisting each of 3 or 4, &c. continued numbers.

Prop. 12. shows, that of the powers of numbers in arithmetical progression, the orders of differences which become equal, are the second differences in the squares, the 3d differences in the cubes, the 4th differences in the 4th powers, &c. And from hence it is shown, how to construct all those powers by the continual addition of their differences: As had been long before more fully explained by Briggs.

In the next, or 13th prop. our author explains his compendious method of raising the tables of logarithms, showing how to construct the logarithms by addition only, from the properties contained in the 8th, 9th, and 12th propositions. For this purpose he makes use of the quantity $\backslash frac\{a\}\{b-c\}$, which by division he

resolves into this infinite series $\backslash frac\{a\}\{b\}\; +\; \backslash frac\{ac\}\{bb\}\; +\; \backslash frac\{ac^2\}\{b^3\}\; +\; \backslash frac\{ac^3\}\{b^4\}$, &c. (in infinitum). Putting then $a\; =\; 100$ the arithmetical mean between the terms of the ratio $\backslash frac\{1\}\{100\}$, $b\; =\; 100000$, and $c$ successively equal to 0.5, 1.5, 2.5, &c. that so $b-c$ may be respectively equal to 99999.5, 99998.5, 99997.5, &c. the corresponding means between the terms of the ratios $\backslash frac\{1\}\{100000\},\; \backslash frac\{1\}\{1000000\},\; \backslash frac\{1\}\{10000000\}$, &c. it is evident that $\backslash frac\{a\}\{b-c\}$ will be the quotient of the 2d term divided by the 1st in the proportions mentioned in the 8th and 9th propositions; and when each of these quotients are found, it remains then only to multiply them by the constant 3d term 43429, or rather 43429.8, of the proportion, to produce the logarithms of the ratios $\backslash frac\{1\}\{100000\},\; \backslash frac\{1\}\{1000000\},\; \backslash frac\{1\}\{10000000\}$, &c. till $\backslash frac\{1\}\{100000000\}$; then adding these continually to 4 the logarithm of 10000 the least number, or subtracting them from 5 the logarithm of the highest term 100000, there will result the logarithms of all the absolute numbers from 10000 to 100000. Now when $c$ is = 0.5, then

&c. But instead of constructing all the values of $\backslash frac\{a\}\{b-c\}$ in the usual way of raising the powers, he directs them to be found by addition only, as in the last proposition. Having thus found all the values of $\backslash frac\{a\}\{b-c\}$, the author then shows, that they may be drawn into the constant logarithm 43429 by addition only, by the help of the annexed table of the first 9 products of it.

The author then distinguishes which of the logarithms it may be proper to find in this way, and which from their component parts. Of these the logarithms of all even numbers need not be thus computed, being composed from the number 2; which cuts off one half of the numbers: neither are those numbers to be computed which end in 5, because 5 is one of their factors; these last are $\backslash frac\{1\}\{10\}$ of the numbers; and the two together $\backslash frac\{1\}\{2\}\; +\; \backslash frac\{1\}\{10\}$ make $\backslash frac\{1\}\{5\}$ of the whole: and of the other $\backslash frac\{1\}\{5\}$, the

$\backslash frac\{1\}\{5\}$ of them, or $\backslash frac\{1\}\{2\}$ of the whole, are composed of 3; and hence $\backslash frac\{1\}\{2\}\; +\; \backslash frac\{1\}\{5\}$, or $\backslash frac\{1\}\{10\}$ of the numbers, are made up of such as are composed of 2, 3, and 5. As to the other numbers which may be composed of 7, of 11, &c.; he recommends to find their logarithms in the general way, the same as if they were incompites, as it is not worth while to separate them in so easy a mode of calculation. So that of the 90 chiliads of numbers from 10000 to 100000, only 24 chiliads are to be computed. Neither indeed are all of these to be calculated from the foregoing series for $\backslash frac\{a\}\{b-c\}$, but only a few of them in that way, and the rest by the proportion in the 8th proposition. Thus, having computed the logarithms of 10003 and 10013, omitting 10023 as being divisible by 3, estimate the logarithms of 10033 and 10043, which are the 30th numbers from 10003 and 10013; and again, omitting 10053, a multiple of 3, find the logarithms of 10063 and 10073. Then by prop. 8.

As 10048, the arithmetical mean between 10033 and 10063, to 10018, the arithmetical mean between 10003 and 10033, to 13006, the difference between the logarithms of 10003 and 10033, to 12967, the difference between the logarithms of 10033 and 10063;

That is, 1st As

Again, As

And 3dly, As

And with this our author concludes his compendium for constructing the tables of logarithms.

§ 6. Gregory's Method.

This is founded upon an analogy between a scale of logarithmic tangents and Wright's protraction of the nautical meridian line consisting of the sums of the secants. It is not known by whom this discovery was made; but, about 1645, it was published by Mr Henry Bond, who mentions this property in Norwood's Epitome of Navigation. The mathematical demonstration of it was first investigated by Mercator; who, with a view to make some advantage of his discovery, offered, in the Philosophical Transactions for June 4th 1666, to lay a wager with any one concerning it; but this proposal not being accepted, the demonstration was not published. Other mathematicians, however, soon found out the mystery; and in two years after, Dr Gregory published a demonstration, and from this and other similar properties he showed a method of computing the logarithmic fines and tangents by means of an infinite series. Several of these were invented by him, and the method of applying them laid down by himself and others; but Mr Hutton thinks that a shorter and better method than any they proposed

might have been found by computing, by means of the series, only a few logarithms of small ratios, in which the terms of the series would have decreased by the powers of 10 or some greater number, the numerators of all the terms being unity, and their denominators the powers of 10 or some greater number, and then employing these few logarithms, so computed, to the finding of the logarithms of other and greater ratios by the easy operations of mere addition and subtraction. This might have been done for the logs. of the ratios of the first ten numbers, 2, 3, 4, 5, 6, 7, 8, 9, 10, and 11, to 1, in the following manner, communicated by Mr Baron Maferes.—In the first place the logarithm of the ratio of 10 to 9, or of 1 to $\backslash frac\{9\}\{10\}$, or of 1 to $1\; -\; \backslash frac\{1\}\{10\}$, is equal to the series In like manner are easily found the logarithms of the ratios of 11 to 10; and then by the same series those of 121 to 120, and of 81 to 80, and of 2401 to 2400; in all which cases the series would converge still faster than in the two first cases. We may then proceed by mere addition and subtraction of logarithms, as follows.

S 2

Having

Having thus got the logarithm of the ratio of 2 to 1, or, in common language, the logarithm of 2, the logarithms of all sorts of even numbers may be derived

from those of the odd numbers which are their coefficients with 2 or its powers. We may then proceed as follows.

Thus we have got the logarithms of 2, 3, 4, 5, 6, 7, 8, 9, 10, and 11. And this is upon the whole, perhaps, the best method of computing logarithms that can be taken.—This method of computing logarithms is very nearly the same with that of Sir Isaac Newton in his second letter to Mr Oldenburg, dated October 1676.

§ 7. Construction of Logarithms by Fluxions.

FROM the definition and description of logarithms given by Napier, and of which we have already taken notice, it appears that the fluxion of his, or the hyperbolic logarithm of any number, is a fourth proportional to that number, its logarithm and unity; or, which is the same, that it is equal to the fluxion of the number divided by the number: For the description shows that $z\; :\; za$ or $1\; :\; z$ the fluxion of $z\; :\; za$, which therefore is $\backslash frac\{z\}\{1\}$; but $za$ is also equal to the fluxion of the logarithm $A$, &c. by the description; therefore the fluxion of the logarithm is equal to $\backslash frac\{z\}\{1\}$, the fluxion of the quantity divided by the quantity itself. The same thing appears again at art. 2. of of that little piece in the appendix to his Constitutio Logarithmorum, intitled Habitudines Logarithmorum & suorum naturalium numerorum invicem; where he observes, that as any greater quantity is to a less, so is the velocity of the increment or decrement of the logarithms at the place of the less quantity to that at the greater. Now this velocity of the increment or decrement of the logarithms being the same thing as their fluxions, that proportion is this $x\; :\; a\; :\; :\; \backslash text\{flux.\; log.\; \}\; a\; :\; \backslash text\{flux.\; log.\; \}\; x$; hence if $a$ be $=1$, as at the beginning of the table of numbers, where the fluxion of the logs. is the index or characteristic $c$, which is also one in Napier's or the hyperbolic logarithms, and 43429, &c. in Briggs's, the same proportion becomes $x\; :\; 1\; :\; :\; c\; :\; \backslash text\{flux.\; log.\; \}\; x$; but the constant fluxion of the numbers is also 1, and therefore that proportion is also this $x\; :\; x\; :\; :\; c\; :\; \backslash frac\{cx\}\{x\}\; =$ the fluxion of the logarithm of $x$; and in the hyperbolic logarithms, where $c$ is $=1$, it becomes $\backslash frac\{x\}\{x\}\; =$ the fluxion of Napier's or the hyperbolic logarithm of $x$. This same property has also been noticed by many other

authors since Napier's time. And the same or a similar property is evidently true in all the systems of logarithms whatever, namely, that the modulus of the system is to any number as the fluxion of its logarithm is to the fluxion of the number.

Now from this property, by means of the doctrine of fluxions, are derived other ways for making logarithms, which have been illustrated by many writers on this branch; as Craig, Jo. Bernoulli, and almost all the writers on fluxions. And this method chiefly consists in expanding the reciprocal of the given quantity in an infinite series, then multiplying each term by the fluxion of the said quantity, and lastly taking the fluents of the terms; by which there arises an infinite series of terms for the logarithm sought. So, to find the logarithm of any number $N$, put any compound quantity for $N$, as suppose $\backslash frac\{n+x\}\{n\}$; then the flux.

of the log. or $\backslash frac\{N\}\{n\}$ being $\backslash frac\{x\}\{n+x\}\; =\; \backslash frac\{x\}\{n\}\; -\; \backslash frac\{x^2\}\{n^2\}\; +\; \backslash frac\{x^3\}\{n^3\}\; -\; \backslash frac\{x^4\}\{n^4\}$, &c.

the fluents give log. of $N$ or log. of $\backslash frac\{n+x\}\{n\}\; =\; \backslash frac\{x\}\{n\}\; -\; \backslash frac\{x^2\}\{2n^2\}\; +\; \backslash frac\{x^3\}\{3n^3\}\; -\; \backslash frac\{x^4\}\{4n^4\}$, &c. And writing $-x$ for $x$ gives log.

$\backslash frac\{n-x\}\{n\}\; =\; -\backslash frac\{x\}\{n\}\; +\; \backslash frac\{x^2\}\{2n^2\}\; -\; \backslash frac\{x^3\}\{3n^3\}\; +\; \backslash frac\{x^4\}\{4n^4\}$, &c. Also, because

$\backslash frac\{n\}\{n+x\}\; =\; 1\; -\; \backslash frac\{x\}\{n\}$, or log. $\backslash frac\{n\}\{n+x\}\; =\; 0\; -\; \backslash text\{log.\; \}\; \backslash frac\{n+x\}\{n\}$, we

have log. $\backslash frac\{n\}\{n+x\}\; =\; -\backslash frac\{x\}\{n\}\; +\; \backslash frac\{x^2\}\{2n^2\}\; -\; \backslash frac\{x^3\}\{3n^3\}\; +\; \backslash frac\{x^4\}\{4n^4\}$, &c. and log.

$\backslash frac\{n\}\{n-x\}\; =\; +\backslash frac\{x\}\{n\}\; +\; \backslash frac\{x^2\}\{2n^2\}\; +\; \backslash frac\{x^3\}\{3n^3\}\; +\; \backslash frac\{x^4\}\{4n^4\}$, &c.

And by adding and subtracting any of these series to or from one another, and multiplying or dividing their corresponding numbers, various other series for logarithms may be found, converging much quicker than these do.

In like manner, by assuming quantities otherwise compounded for the value of $N$, various other forms of logarithmic series may be found by the same means.

§ 8. Mr Long's Method.

THIS method was published in the 339th number of the Philosophical Transactions; and is performed by means of a small table containing eight classes of logarithms, as follows.

Here, because the logarithms in each class are the continual multiples 1, 2, 3, &c. of the lowest, it is evident that the natural numbers are so many scales of geometrical proportions, the lowest being the common ratio, or the ascending numbers are the 1, 2, 3, &c. powers of the lowest, as expressed by the figures 1, 2, 3, &c. of their corresponding logarithms. Also the last number in the first, second, third, &c. class, is the 10th, 100th, 1000th, &c. root of 10; and any number in any class is the 10th power of the corresponding number in the next following class.

To find the logarithm of any number, as suppose of 2000, by this table: Look in the first class for the number next less than the first figure 2, and it is 1,995,262,315, against which is 3 for the first figure of the logarithm sought. Again, dividing 2, the number proposed, by 1,995,262,315, the number found in the table, the quotient is 1,002,374,467; which being looked for in the second class of the table, and finding neither its equal nor a less, 0 is therefore to be taken for the second figure of the logarithm; and the same quotient 1,002,374,467 being looked for in the third class, the next less is there found to be 1,002,305,238, against which is 1 for the third figure of the logarithm; and dividing the quotient 1,002,374,467 by the said next less number 1,002,305,238, the new quotient is 1,000,069,270; which being sought in the fourth class gives 0, but sought in the fifth class gives 2, which are the fourth and fifth figures of the logarithm sought: again, dividing the last quotient by 1,000,069,270, the next less number in the table, the quotient is 1,000,023,015, which gives 9 in the 6th class for the 6th figure of the logarithm sought: and again dividing the last quotient by 1,000,023,015, the next less number, the quotient is 1,000,002,291, the next less than which in the 7th class gives 9 for the 7th figure of the logarithm: and dividing the last quotient by 1,000,002,291, the quotient is 1,000,000,219, which gives 9 in the 8th class for the 8th figure of the logarithm: and again the last quotient 1,000,000,219 being divided by 1,000,000,207 the next less, the quotient 1,000,000,012 gives 5 in the same 8th class, when one figure is cut off, for the 9th figure of the logarithm sought. All

which figures collected together give 3,301,029,995 for Briggs's logarithm of 2000, the index 3 being supplied; which logarithm is true in the last figure.

To find the number answering to any given logarithm, as suppose to 3,301,0300: omitting the characteristic, against the other figures 3, 0, 1, 0, 3, 0, 0, as in the first column in the margin, are the several numbers as in the second column, found from their respective 1st, 2d, 3d, &c. classes; the effective numbers of which multiplied continually together, the last product is 2,000,000,199,66, which, because the characteristic is three, gives 2000,000,199,66 or 2000 only for the required number answering to the given logarithm.

§ 9. Mr Hutton's Practical Rule for the Construction of Logarithms.

THE methods laid down in the above sections are abundantly sufficient to show the various principles upon which logarithms may be constructed; though there are still a variety of others which our limits will not admit of our inserting: The following rule is added from Mr Hutton's Treatise on the subject, for the sake of those who do not choose to enter deeply into these investigations.

Call $z$ the sum of any number whose logarithm is sought, and the number next less by unity; divide $z$ by 2, and reserve the quotient; divide the reserved quotient by the square of $z$, and reserve this quotient; divide this last quotient also by the square of $z$, and again reserve this quotient; and thus proceed continually, dividing the last quotient by the square of $z$ as long as division can be made. Then write these quotients orderly under one another, the first uppermost, and divide them respectively by the uneven numbers 1, 3, 5, 7, 9, 11, &c. as long as division can be made; that is, divide the 1st reserved quotient by 1, the 2d by 3, the 3d by 5, the 4th by 7, &c. Add all these last quotients together, and the sum will be the logarithm of $z$; and therefore to this logarithm add also the logarithm.

Construction of Logarithms. $\backslash overbrace\{\backslash text\{garithm\; of\; \}\; a\; \backslash text\{\; the\; next\; less\; number,\; and\; the\; sum\; will\; be\; the\; required\; logarithm\; of\; \}\; b\; \backslash text\{\; the\; number\; proposed.\}\}$

Ex. 1. To find the Log of 2.—Here the next less number is 1, and $2+1=3=z$, whose square is 9. Then,

Log. $\backslash frac\{1\}\{2\}$ - 301029995
Add L. 1 - 000000000

Log. of 2 - 301029995

Ex. 2. To find the log. of 3.—Here the next less number is 2, and $2+3=5=z$, whose square is 25, to divide by which always multiply by .04. Then

L. $\backslash frac\{1\}\{2\}$ - 176091260
L. 2 add - 301029995

L. 3 - 477121255

Then because the sum of the logarithms of numbers gives the logarithm of their product, and the difference of the logarithms gives the logarithm of the quotient of the numbers, from the above two logarithms, and the logarithm of 10, which is 1, we may raise a great many logarithms, thus:

Ex. 3. Because $2\; \backslash times\; 2\; =\; 4$, therefore

to L. 2 - 301029995$\backslash frac\{1\}\{2\}$
add L. 2 - 301029995$\backslash frac\{1\}\{2\}$

sum is L. 4 - 602059991$\backslash frac\{1\}\{2\}$

Ex. 4. Because $2\; \backslash times\; 3\; =\; 6$, therefore

to L. 2 - 301029995
add L. 3 - 477121255

sum is L. 6 - 778151250

Ex. 5. Because $2^3\; =\; 8$, therefore

L. 2 - 301029995$\backslash frac\{1\}\{2\}$
mult. by - 3

gives L. 8 - 903089987

Ex. 6. Because $3^2\; =\; 9$, therefore

L. 3 - 477121255$\backslash frac\{1\}\{2\}$
mult. by - 2

gives L. 9 - 954242509

Ex. 7. Because $4^2\; =\; 16$, therefore

from L. 10 - 1000000000
take L. 2 - 301029995$\backslash frac\{1\}\{2\}$

leaves L. 5 - 698970004$\backslash frac\{1\}\{2\}$

Ex. 8. Because $12\; =\; 3\; \backslash times\; 4$, therefore

to L. 3 - 477121255
add L. 4 - 602059991

gives L. 12 - 1079181246

And thus by computing, by the general rule, the logarithms of the other prime numbers 7, 11, 13, 17, 19, 23, &c.: and then using composition and division, we may easily find as many logarithms as we please, or may speedily examine any logarithm in the table.

§ 10. Mr Thomas Atkinson of Ballishannon's Method.

In any series of numbers in a geometrical progression, beginning from unity, as in the margin, the series is composed of 0 1 2 3
of continued proportionals, of 1 10 100 100
which the member standing nearest to unity is the common ratio or rate of the proportion. If over or under these another series is placed, as in the example, of numbers in an arithmetical progression, beginning with nought, and whose common difference is unity, the members of this series are called indexes; for they serve to show how many successive multiplications have been made with the common rate to produce that member of the geometrical progression over which each of these indexes does severally stand.

This theory may be considered in another light: If the square root of 10 (that is, of the common rate) is found, it is a mean proportional between 1 and 10, and becomes a new common rate for a new set of continued proportionals, as in the margin. And if the 0.5 1 1.5 2 25
half of unity, which in 1 3.16 10 31.6 100 316.
the former case was the additional difference of the arithmetical progression, is made the additional difference of this new series, and noted as in the example, a new combination is formed of two series agreeing with the first in these remarkable properties, viz. If any two members of the geometrical progression are multiplied together, the sum of their corresponding indexes will become the index of their product; and conversely, if any of them is divided by any other, the difference of their indexes will be found to be the index of the quotient. This theory is indefinite; and repeated extractions may be made with any proposed number of decimals, and bisection made of the corresponding indexes, until one has no more number to work with; and each of the mean proportionals thus found between 1 and 10, will be found a member of every new geometrical progression formed by every smaller root; and consequently all the roots thus found, together with their corresponding indexes, have, in cases of multiplication or division, the same connection as has been just described.

Let those successive roots be found, and noted in the form of a table, and, in another column, let the corresponding indexes found by these directions be regularly

Construc- tion of Logarithms

larly noted, each opposite to its own roots. These indexes are commonly known by the denomination of logarithms; the roots themselves may be called natural numbers.

These roots are composed of natural numbers seldom or never wanted; but from them the logarithms of such as are of general use may be thus found.

Suppose 2 the proposed number, one must examine the table of roots; there he will find 3.16, &c. &c. the nearest to 2 of those which are greater; and 1.778, &c. &c. also nearest to it of those which are less. He

may make a division at his pleasure, either $\backslash frac\{3.16\}\{2\}$ or $\backslash frac\{2\}\{1.77\}$; yet let the choice fall on what will yield the

smallest quotient, and let the circumstances of the calculation be noted, as in the margin, for future direction. Here

.0000087837, . . . 0000038147 its logm. . . . . 0000057968 of the quotient.
.0000025175, its logm.

Thus knowing that 0.0000025175, or such like, is the logm. of the last quotient, one may have that of 2, if he will but call to mind the following circumstances.

In every case of division, if he has logarithms of quotient and divisor, he has also that of the dividend, by adding the two first together: if he has the logarithm of the dividend, and that of either the divisor or quotient, he may find that of the other; for he has only to subtract what he knows from the logarithm of the dividend, the remainder is what he wants: and lastly, that in every division he made, he took one number from the table of roots whose logarithm is known, being noted in the table, and which he made use of as his direction either as a dividend or a divisor: From these circumstances, one may, by the help of the logarithm just found, discover the logarithm of that number of the last division, whether it be dividend or divisor, which was the quotient of the preceding division; and thus, tracing his own work backwards by his notes from quotient to quotient, be they ever so few or ever so many, he will come at last by addition and subtraction to the logarithm of the proposed number.

By this method, the logarithm of any number within the compass of the table of roots may be found: if a greater is proposed, suppose 9.495, it must be made 9.495, and its logarithm found; then it must be re-

—3.301029995664 the logarithm of the fraction given.
7 the power to which it is to be raised.

—19.107209969648 the logarithm of the answer.

This differs from the like work in whole numbers only in this, that, in multiplying the decimals, one has at last 2 to be carried from them to the whole numbers; this is to be considered as +2, then $-3\; \backslash times\; 7\; =\; -21$, and $-21\; +\; 2\; =\; -19$ to be noted the index of the answer. Extraction of the roots is only the converse of this. Suppose —19.107209969648 given, to find that root whose exponent number is 7.

Construc- tion of Logarithms

$\backslash frac\{2\}\{1.77\}\; =\; 1.1246$. With this quotient let the table be applied to as before, and 1.1246, &c. will be found to be between 1.154, &c. &c. and 1.074, &c. &c. and division to be made as in the example. In this manner one is to proceed with each successive quotient, till at length he has one in which the number of the initial decimal noughts is equal at least, if not greater than that of the significant figures. Here the work of division may be discontinued; and as it will rarely happen, that one will not have an additional initial nought for every division, the number cannot be great in calculations of a moderate extent. Having at last found a quotient such as was described, and supposing the number of decimals to be 10, one may readily find the logm. of that quotient thus:—Suppose the quotient 1.0000057968; he is to look into the table of roots for those noted with 5 initial decimal noughts, and from any one of these and its corresponding logm. state thus:

stored to the proposed form, and have a proper index noted before the decimals just found. How to do this is too well known to have occasion to mention it here.

The reason for finding the logarithm of the last quotient by the common proportion is this: He who has made a table of roots, will find, by inspection only, that as initial noughts come into the decimal parts of the roots, the significant figures just immediately following them do assume the form of a geometrical progression, descending, whose common rate or divisor is 2, as is just the case with the whole of the decimals of the corresponding logarithms; and that the number of the significant figures endowed with this property is generally equal to that of the initial noughts: so far as this, and no farther, the common proportion will hold between the significant figures of the decimals in the roots and the same number of places in the logarithms; and for this reason it was needful to continue the successive divisions till a quotient was found so circumstanced, that its logarithm could be found by the proportion.

The same gentleman hath also favoured us with the following new method

Of extracting Roots of Fractions by LOGARITHMS.

The easiest way to explain this, is first to give an example of involving such numbers.

As 7 is the exponent number here, one may in his mind multiply it by 2 for a trial, as in common division; but the product = 14 being less than 19, must be rejected; then he may try it with 3, this yields 21 for a product. This 3 must be noted with —3.301029995664 a negative sign for the index of the new logarithm. Then, on comparing 19 with 21, the difference is

2. This 2 must be carried as 20 to the decimals, and one must from that carry on the division of the de-

cimals with 7 for a divisor, as is usually done in other cases.

Another EXAMPLE.

Suppose —1.4771212545 given, to extract the root of its 5th power.
—1.8954232109 the logarithm of the root.

For 5, the exponent of the root $\backslash times\; 1$ is greater than the index of the given logarithm, and 4 is the remainder. Then —1 becomes the index of the logarithm of the root; and 4 = the overplus, is to be carried as 40 to the decimals; and from that, division is to be made with 5 as a divisor for the rest of the work.

SECT. III. Explanation and Use of the Table, with a general Account of the various Sciences to which Logarithms may be applied.

§ 1. To find by the table the Logarithm of any number.

If the number be under 100, it is easily found in the first division at the head of the first page; if it be betwixt 100 and 1000, over against the number in the first column of the following pages, in the next column under 0 will be found the logarithm required. If the number be betwixt 1000 and 10000, the first three figures of the number are to be found in the column marked N° and the fourth figure at the top, and in the column under it, lineally against the first three figures, will be found the logarithm required, changing the index 2 into 3. The column marked Diff. and showing the common difference by which each of these columns increases, serves to find the logarithms of numbers beyond 10000. Thus,

To find the logarithm for a number greater than any in the common canon, but less than 10000000. — Cut off four figures on the left of the given number, and seek the logarithm in the table; add as many unites to the index as there are figures remaining on the right; subtract the logarithm found from the next following it in the table; then, as the difference of numbers in the canon is to the tabular distance of the logarithms answering to them, so are the remaining figures of the given number to the logarithmic difference; which, if it be added to the logarithm before found, the sum will be the logarithm required. Suppose v. gr. the logarithm of the number 92375 required. Cut off the four figures 9237, and to the characteristic of the logarithm corresponding to them, add an unit; then,

From the logarithm. of the numb. 9238 = 3.965578

Subtract logarithm. numb. — 9237 = 3.965531

Remains tabular difference 47

Then 10 : 47 :: 5 : 23

Now to the logarithm — 4.965531

Add the difference found — 23

The sum is the logarithm required. — 4.965554

Or more briefly; find the logarithm of the first four figures as before; then multiply the common difference which stands against it by the remaining figures of the given number; from the product, cut off as many figures at the right hand as you multiplied by, and add the remainder to the logarithm before found, fitting it with a proper index. Thus $47\; \backslash times\; 5\; =\; 235$; cut off 5 and add 23.

To find the logarithm of a fraction. — Subtract the logarithm of the numerator from that of the denominator, and to the remainder prefix the sign of sub-

traction. Thus suppose it required to find the logarithm of the fraction $\backslash frac\{1\}\{7\}$,

Logarithm of 7 = 0.845098

Logarithm of 3 = 0.477121

Logarithm of $\backslash frac\{1\}\{7\}$ = —0.367977

The reason of the rule is, that a fraction being the quotient of the numerator divided by the denominator, its logarithm must be the difference of the logarithms of those two; so that the numerator being subtracted from the denominator, the difference becomes negative. Stifelius observed, that the logarithms of a proper fraction must always be negative, if that of unity be 0; which is evident, a fraction being less than one.

Or, the logarithm of the denominator, though greater than that of the numerator, as in the case of a proper fraction, may be subtracted from it, regard being had to the sign of the index, which alone in that case is negative. Thus,

Log. of 3 = 0.477121

Log. of 7 = 0.845098

Log. of $\backslash frac\{1\}\{7\}$ = 1.632023 which produces the same effect in any operation as that before found, viz. —0.367977, this being to be subtracted, and the other to be added.

Or again, the fraction may be reduced to a decimal, and its logarithm found; which differs from that of a whole number only in the index, which is to be negative.

For an improper fraction v. gr. $\backslash frac\{3\}\{5\}$, its numerator being greater than its denominator, its logarithm is had by subtracting the logarithm of the latter from that of the former.

The logarithm of 9 = 0.9542425

Logarithm of 5 = 0.6989700

Logarithm $\backslash frac\{3\}\{5\}$ = 0.2552725

In the same manner may a logarithm of a mixed number, as $3\backslash frac\{1\}\{2\}$, be found, it being first reduced into an improper fraction $\backslash frac\{7\}\{2\}$.

Or, this improper fraction may be reduced to a mixed number, whose logarithm must be found as if it were wholly integral, and its index taken according to the integral part. We shall here observe, that the logarithms of whole numbers are added, subtracted, &c. according to the rules of these operations in decimal arithmetic; but with regard to the management of logarithms with negative indices, the same rules are to be observed as those given in algebra for like and unlike signs.

In addition, all the figures except the index, are reckoned positive, and therefore the figure to be carried to the index from the other part of the logarithm takes away so much from the negative index. Thus $1.8683326\; +\; 3.698072\; =\; 1.5622298$. In subtraction, if either one or both of the logarithms have negative indices, you must change the sign of the index of the sub-

subtrahend, after you have carried to it what may arise from the decimal part, and then add the indices: thus $1.562298\; -\; 1.863326\; =\; 3.698972$. In multiplication, what is carried from the product of the other parts of the logarithms must be subtracted from the product of the indices: thus $2.477121\; \backslash times\; 5\; =\; 8.385605$. In division, if the divisor will exactly measure the index, proceed as in common arithmetic; e. g. $4.924782\; \backslash div\; 2\; =\; 2.462391$. But if the divisor will not exactly measure the index, add units to the index, till you can exactly divide it, and carry these units to the next number: e. g. $8.385605\; \backslash div\; 5\; =\; 2.477121$.

To find the number corresponding to any given logarithm.—If the logarithm be within the limits of the table, i. e. if its index does not exceed 3, then neglecting the index, look down in the column of logarithms under 0, for the two or three first figures of your given logarithm; and if you exactly find all the figures of the given logarithm in that column, you have the number corresponding at the left hand: But if you do not find your logarithm exactly in the column under 0, you must run through the other columns till you find it exactly, or till you obtain the next least logarithm; and in the column of numbers lineally against it, you have the first 3 figures of the number sought, to which join the figure over the column, where your logarithm or its next least was found, and you have the corresponding number, e. g. the number answering to the logarithm 3.544812 is 3506.

If the index of this logarithm had been 1, then the two last figures of the number would have been decimal; with the index 0, its corresponding number would have been 3.506; with 1, .3506; with 2, .03506, &c.

If the logarithm cannot be found exactly, take the next least, and make the difference between the given logarithm and the next least the numerator of a fraction whose denominator shall be the common difference, and add the fraction to the number found in the table.

To find the number corresponding to a logarithm greater than any in the table.—First, from the given logarithm, subtract the logarithm of 10, or 100, or 1000, or 10,000, till you have a logarithm that will come within the compass of the table; find the number corresponding to this, and multiply it by 10, or 100, or 1000, or 10,000, the product is the number required.

Suppose, for instance, the number corresponding to the logarithm 7.7589982 be required: subtract the logarithm of the number 10,000, which is 4.0000000, from 7.7589982; the remainder is 3.7589982, the number corresponding to which is 5741 $\backslash frac\{1\}\{100\}$: this multiplied by 10,000, the product is 5741100, the number required.

Otherwise seek the decimal member of the logarithm in the table, and if you can find it exactly, you have the four first figures of the number in the table, to which affix as many ciphers as the given index exceeds 3, and it is the number required. But if you cannot find the logarithm exactly, take the next least, and find the four first figures of the corresponding number; then take the difference betwixt the given logarithm and the next least, and annex to it as many ciphers as the index exceeds 3; then divide by the common dif-

ference, and affix the quotient to the four first figures, and you have the number required.

To find the number corresponding to a negative logarithm.—To the given negative logarithm add the last logarithm of the table, or that of the number 10000; i. e. subtract the first from the second, and find the number corresponding to the remainder; this will be the numerator of the fraction, whose denominator will be 10000; e. g. suppose it to be required to find the fraction corresponding to the negative logarithm

The remainder is — 3.6320233, the number corresponding to which is 4285 $\backslash frac\{1\}\{100\}$, the fraction sought therefore is $\backslash frac\{4285\}\{100\}$. The reason of the rule is, that as a fraction is the quotient, arising on the division of the numerator by the denominator, unity will be to the fraction as the denominator to the numerator; but as unity is to the fraction corresponding to the given negative logarithm, so is 10000 to the number corresponding to the remainder: therefore, if 10000 be taken for the denominator, the number will be the numerator of the fraction required.

The negative logarithm —0.367977 is equal to the logarithm 1.632023, and the number answering to it, found in the manner already directed, will be 4285 $\backslash frac\{1\}\{100\}$.

The fines, tangents, &c. of any arch are easily found by seeking the degree at the top, if the arch be less than 45°, and the minutes at the side, beginning from the top, and by seeking the degree, &c. at the bottom, if the arch is greater than 45°. If a given logarithmic sine or tangent falls between those in the tables, then the corresponding degrees and minutes may be reckoned $\backslash frac\{1\}\{4\}$, $\backslash frac\{1\}\{2\}$, or $\backslash frac\{3\}\{4\}$, &c. minutes more than those belonging to the nearest less logarithm in the tables, according as its difference from the given one is $\backslash frac\{1\}\{4\}$, or $\backslash frac\{1\}\{2\}$, or $\backslash frac\{3\}\{4\}$, &c. of the difference between the logarithm next greater and next less than the given log.

§ 2. Of the various Sciences to which Logarithms may be applied.

As these artificial numbers constitute a new species of arithmetic capable of performing every thing which can be done in the old way, it is plain that its use must be equally extensive, and that in every science in which common arithmetic can be useful, the logarithmical arithmetic must be much more so, by reason of its being more easily performed. Though the general principles of logarithmical arithmetic have been already laid down, we shall here, in order to render the subject still more plain, subjoin the following practical rules.

I. Multiplication by Logarithms.

Add together the logarithms of all the factors, and the sum is a logarithm, the natural number corresponding to which will be the product required.

Observing to add, to the sum of the affirmative indices, what is carried from the sum of the decimal parts of the logarithms.

And that the difference betwixt the affirmative and negative indices is to be taken for the index to the logarithm of the product.

Div. from by Ex. 1. To multiply 23.14 by 5.062.
23.14 its log. is 1.3643634
5.062 its log. is 0.7043221

Product 117.1347 - 2.0686855

Ex. 2. To mult. 2.581926 by 3.457291.
2.581926 its log. is 0.4119438
3.457291 - - 0.5387359

Prod. 8.92647 - 0.9506797

Ex. 3. To mult. 3.902, and 597.16, and .0314728 all together.

3.902 its log. is 0.5912873
597.16 - - 2.7760907
.0314728 - - 2.4979353

Prod. 73.33533 - 1.8653133

The 2 cancels the 2, and the 1 to carry from the decimals is set down.

Ex. 4. To mult. 35.86, and 2.1046, and 0.8372, and 0.0294 all together.

35.86 its log. is 0.5546103
2.1046 - - 0.3231696
0.8372 - - 1.9228292
0.0294 - - 2.4683473

Prod. 1857618 - 1.2689564

Here the 2 to carry cancels the 2, and there remain the 7 to set down.

II. Division by Logarithms.

From the logarithm of the dividend subtract the logarithm of the divisor, the remainder is a logarithm whose corresponding number will be the quotient required.

But first observe to change the sign of the index of the logarithm of the divisor, viz. from negative to affirmative, or from affirmative to negative; then take the sum of the indices if they be of the same kind, or their difference when of different signs, with the sign of the greater, for the index to the logarithm of the quotient.

And when 1 is borrowed in the left-hand place of the decimal part of the logarithm, add it to the index of the logarithm of the divisor when that index is affirmative, but subtract it when negative; then let the index thus found be changed, and worked with as before.

Ex. 1. To divide 24163 by 4567.

Divide 24163 its log. 4.3831509
Divis. 4567 - - 3.6596310

Quot. 5.290782 - 0.7235199

Ex. 2. To divide 37.149 by 523.76.

Divis. 37.149 its log. 1.5699471
Divis. 523.76 - - 2.7191323

Quot. 0.7092752 - 2.8508148

Ex. 3. To divide .06314 by .007241.

Divis. .06314 its log. 2.8003046
Divis. .007241 - - 3.8597985

Quot. 8.719792 - 0.9405061

Here 1 carried from the decimals to the 3 makes it become 2, which taken from the other 2, leaves 0 remaining. Rule of Theory Logarithms

Ex. 4. To divide .7438 by 12.9476.

Divis. .7438 its log. 1.8714562
Divis. 12.9476 - - 1.1121893

Quot. 0.5744694 - 2.7592669

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

III. The Rule of Three, or Proportion.

Add the logarithms of the 2d and 3d terms together, and from their sum subtract the logarithm of the 1st by the foregoing rules; the remainder will be the logarithm of the 4th term required.

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

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

Ex. 1. To find a 4th proportional to 72.34, and 2.519, and 357.4862.

As 72.34 - comp. log. 8.1406215
To 2.519 - - 0.4012282
So 357.4862 - - 2.5532592

To 12.44827 - - 1.0951089

Ex. 2. To find a 3d proportional to 12.796 and 3.24718.

As 12.796 - comp. log. 8.8929258
To 3.24718 - - 0.5115064
So 3.24718 - - 0.5115064

To .8240216 - - 1.9159386

Ex. 3. To find a number in proportion to .379145 as .85132 is to .0649.

As .0649 - comp. log. 11.1877553
To .85132 - - 1.9300928
So .379145 - - 1.5788054

To 4.973401 - - 0.6966535

Ex. 4. If the interest of 100 l. for a year or 365 days be 4.5 l. what will be the interest of 279.25 l. for 274 days?

As { 100 comp. log. { 8.0000000
{ 365 { 7.4377071
To { 279.25 - - 2.4459932
{ 274 { 2.4377506
So 4.5 - - 0.6532125

To 9.433296 - - 0.9746634

Multiply the logarithm of the number given by the proposed index of the power, and the product will be the logarithm of the power sought.

Note. In multiplying a logarithm with a negative index by any affirmative number, the product will be negative.---But what is to be carried from the decimal part of the logarithm will be affirmative.---Therefore the difference will be the index of the product; and is to be accounted of the same kind with the greater.

Ex. 1. To find the 2d power of 2.5791.

Power 6.651756 - - 0.8229364

Ex. 2. To find the cube of 3.07146.

Power 28.97575 - - 1.4620347

Ex. 3. To find the 4th power of .09163.

Power .0000704938 - - 5.8481508

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

Ex. 4. To find the 365th power of 1.0045.

97495
116994
58497
Power 5.148888 - - 0.7117135

Divide the logarithm of the power or given number by its index, and the quotient will be the logarithm of the root required.

Note. When the index of the logarithm is negative, and the divisor is not exactly contained in it without a remainder, increase it by such a number as will

make it exactly divisible; and carry the units borrowed, as so many tens, to the left-hand place of the decimal part of the logarithm; then divide the results by the index of the root.

Ex. 1. To find the square root of 365.

Ex. 2. To find the cube root of 12345.

Ex. 3. To find the 10th root of 2.

Ex. 4. To find the 365th root of 1.045.

Ex. 5. To find the square root of .093.

Here the divisor 2 is contained exactly once in 2 the negative index; therefore the index of the quotient is 1.

Ex. 6. To find the cube root of .00048.

Here the divisor 3 not being exactly contained in 4, augment it by 2, to make it become 6, in which the divisor is contained just 2 times; and the 2 borrowed being carried to the other figures 6, &c. makes 2.6812412, which divided by 3 gives .8937471.

In trigonometry, the use of logarithmical sines, tangents, &c. are used as well as the common arithmetical logarithms; and by using them according to the rules above laid down, the operations are shortened to a degree altogether incredible to persons unacquainted with this invention. With equal facility are the problems in astronomy and navigation solved by their means, as well as those of the higher geometry, fluxions, and in short every thing which requires deep and laborious calculation. For the particular application of them to the different sciences, see the articles NAVIGATION, TRIGONOMETRY, &c.