the first consequent line. Fig. 9 is a circular instrument equivalent to the former; consisting of three concentric circles engraved and graduated upon a plate of an inch and a half diameter. Two legs, A and B, proceed from the centre, having right-lined edges in the direction of radii; and are moveable either singly or together. In using the instrument, place one of the edges at the antecedent and the other at the consequent, and fix them at the angle. Move the two legs then together; and, having placed the antecedent leg at any other number, the other will give the consequent one in the like position on the lines. If the line C D happen to lie between the legs, and B be the consequent leg, the number sought will be found one line farther from the centre than it would otherwise have been: and, on the contrary, it will be found one line nearer in the like case, if A be the consequent leg. This instrument,' says Mr. Nicholson, differing from that represented fig. 6, only in its circular form and the advantages resulting from that form, the lines must be taken to succeed each other in the same manner laterally; so that numbers which

fall either within or without the arrangement of circles will be found on such lines of the arrangement as would have occupied the vacant places, if the succession of lines had been indefinitely repeated sidewise. I approve of this construction, as superior to every other which has yet occurred to me, not only in point of convenience, but likewise in the probability of being better executed; because small arcs may be graduated with very great accuracy, by divisions transferred from a larger original. The instrument, fig. 6, may be contained conveniently in a circle of about four inches and a half diameter. The circular instrument is a combination of the Gunter's line and the sector, with the improvements here pointed out. The property of the sector may be useful in magnifying the differences of the logarithms in the upper part of the line of sines, the middle of the tangents, and the beginning of the versed sines. It is even possible, as mathematicians will easily conceive, to draw spirals, on which graduations of parts, every where equal to each other, will show the ratios of those lines by moveable radii, similar to those in this instrument.'

GENERAL DEFINITION.

1. LOGARITHMS. (logarithme, Fr. Xoyog and apieμoc)-Logarithms, which are the indexes of the ratios of numbers one to another, were first invented by Napier lord Merchiston, a Scottish baron, and afterwards completed by Mr. Briggs, Savilian professor at Oxford. They are a series of artificial numbers, contrived for expediting calculations; and proceed in an arithmetical proportion, as the numbers they answer to do in a geometrical one. The addition and subtraction of logarithms answer to the multiplication and division of the numbers they correspond with; and this saves an infinite deal of trouble. In like manner will the extraction of roots be performed, by bisecting the logarithms of any numbers for the square root, and trisecting them for the cube, and so on. Harris.

The doctrine of logarithms being of great importance, in the science of mathematics, we shall explain their nature and properties more fully in the following section:

SECT. I.-OF THE NATURE AND PROPERTIES OF LOGARITHMS.

2. Let there be two series of numbers, the one constituting an arithmetical progression, and the other a geometrical progression, as follows:

Arith. prog. 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, &c. Geom. prog. 1, 2, 4, 8, 16, 32, 64, 128, 256,512,&c. where the terms stand over each other in such a manner, that 0 in the arithmetical series corresponds to unity in the geometrical series; then we readily perceive, that the two series, so arranged, possess the following properties :

3. I. Let the sum of any two terms of the arithmetical series be taken; and also the product of the corresponding terms of the geometrical series; then, below that term of the arithmetical series, which is equal to the sum, will be "OL. XIII

round a term of the geometrical series equal to the product.

Thus, if the terms of the arithmetical series be 3 and 5, those of the geometrical series will be 8 and 32. Now 3 +58; and 8 × 32=256; and, by inspecting the two series, we find that the term 256 in the geometrical series stands below 8 of the arithmetical series.

4. II. Let the difference of any two terms of the arithmetical series be taken, and also the quotient of the corresponding terms of the geometrical series; then, below that term of the arithmetical series which is equal to the difference, will be found a term of the geometrical series equal to the quotient.

Thus, if the terms of the arithmetical series be 5 and 8, and therefore those of the geometrical series 32 and 256; we shall have 8-5-3, and 256-328; and we find, by inspecting the series, that 8 of the geometrical series stands below 3 of the arithmetical series. This last property is evidently nothing else than the converse of the former.

3. In the preceding geometrical series the common ratio is 2, but it may be any other number whatever, whole or fractional. Thus the same properties will be found to hold true of these series:

Arith. prog. 0, 1, 2, 3, 4, 5, 6, 7, &c. Geom. prog. 1, 3, 9, 27, 81, 243, 729, 2187, &c. where the common ratio of the geometrical series is 3. They also hold true in the following:

Arith. prog. 0, 1, 2, 3, 4, 5, &c. Geom. prog. 1, 3, as, is, is, as, &c. where the common ratio is .

6. To demonstrate that the two foregoing properties must necessarily be true in every case, it is only necessary to write down a geometrical series, according to the algebraic method of notation, thus, ro or 1, r1, r2, m3, gb, q3, pb, p7, &c., where r denotes the ratio of the series, and it

G

presently appears, that the arithmetical series is supplied by the numeral exponents of the terms. Hence it follows, that the properties, which we have ascribed to any geometrical, and its corresponding arithmetical series, are no other than two well known propositions in algebra; namely, that the sum of the exponents of any two powers of an algebraic quantity is equal to the exponent of their product; and that the difference of their exponents is equal to the exponent of their quotient.

7. When the terms of an arithmetical progression are adapted to those of a geometrical progression, as in the three examples given in § 2 and 5, the terms of the arithmetical series are called the logarithms of the corresponding terms of the geometrical series.

Thus, in the first example, 0, 1, 2, 3, &c., are the logarithms of 1, 2, 4, 8, 16, &c., respectively. In the second example, 0, 1, 2, 3, &c., are the logarithms of the numbers 1, 3, 9, 27, &c. And, in the third example, 0, 1, 2, 3, &c., are the logarithms of 1, }, 3, 13, &c.

By applying now the properties of these series which were demonstrated in § 6, to logarithms, and their corresponding numbers, we may define logarithms to be a series of numbers in arithmetical progression, so adapted to another series of numbers in geometrical progression, that the sums and differences of the former correspond to and show the products and quotients of the latter.

8. From this definition of logarithms it appears, that there may be an infinity of different systems according as one or other geometrical series is adapted to the arithmetical series, 0, 1, 2, 3, &c. It may, however, be readily supposed that some systems are better suited to general calculation than others. Accordingly it has been found convenient in practice to adopt that in which the logarithm of 10 is unity: the series of numbers which may have their logarithms expressed by integers being as in the following table:

Logarithms 0, 1, 2, 3, 4, &c. Numbers 1, 10, 100, 1000, 10,000, &c. 9. With respect to the numbers between 1 and 10, 10 and 100, &c., and their corresponding logarithms, they may be understood to be supplied by interpolation, thus: Conceive a great number of geometrical proportionals to be inserted between the natural numbers 1 and 10 which are the extremes; also an equal number of arithmetical proportionals between their logarithms 0 and 1 then, if the number of geometricai proportionals be sufficiently great, some one or other of them will be sufficiently near to each of the natural numbers 2, 3, 4, 5, &c., to 9, as to admit of the one being taken for the other, without any sensible error. There will also be a corresponding logarithm to each, which, as it will be less than unity, may be most conveniently expressed by a decimal fraction.

10. Let us suppose the number of geometrical proportionals between 1 and 10, and also the number of arithmetical proportionals between O and 1, co be 9999, and therefore the number of terms, including 1 and 10, 10001. Then the 3011th term of the geometrical series will be

1.9999, or 2 nearly, and the corresponding term of the arithmetical series, 3010: therefore the logarithm of 2 is 3010, nearly. Again, the 4772d term of the geometrical series will be 2.9999, or 3 nearly; and the corresponding term of the arithmetical series 4771; therefore the logarithm of 3 is 4771 nearly; and so on with respect to other numbers.

11. If we suppose the series of natural numbers 1, 2, 3, 4, 5, &c., to be arranged in a table, so that each number may stand opposite to its corresponding logarithm; it is evident from the properties which we have shown to belong to logarithms, that, by means of such a table, the arithmetical operations of multiplication, division, involution, and evolution, may be performed with great facility.

12. For, since the sum of the logarithms of any two numbers is equal to the logarithm of their product, § 3, the product of any two numbers will be found in the table opposite to that logarithm which is the sum of the logarithms of the numbers.

Again, because the difference of the logarithms of two numbers is equal to the logarithm of the quotient arising from the division of the one number by the other, § 4, that quotient will be found in the table, opposite to the logarithm which is excess of the logarithm of the dividend, above that of the divisor.

13. Involution is performed by multiplying the root into itself a number of times, which is one less than the exponent of the power; therefore, if the logarithm of the root be multiplied by the exponent of that power, the product will be the logarithm of the power of the root. And, evolution being the reverse of involution, the logarithm of any root of a number will be had if we divide the logarithm of that number by the index of the root; and thence the root itself may be found, by inspecting the table of logarithms.

14. Upon the whole, therefore, it appears that by means of a table containing the series of natural numbers, 1, 2, 3, &c., as far as may be convenient, and their corresponding logarithms, the operations of multiplication and division may be reduced to the more simple operations of addition and subtraction; and the operations of involution and evolution, to those of multiplication and division.

SECT. II.-HISTORY OF LOGARITHMS.

15. The properties of a geometrical series, which constitute the foundation of the doctrine of logarithms, appear to have been known as far back as the days of Archimedes; for that celebrated mathematician makes use of them in his work entitled Arenarius, or Treatise on the Number of Sands. The same properties are also mentioned in the writings of Stifelius, a German mathematician, who lived about the middle of the sixteenth century. It does not, however, appear, that any person perceived all the advantages which might be derived from these properties, till about the beginning of the seventeenth century, when their utility was rendered evident by the happy invention of logarithms.

16. This discovery, certainly one of the most valuable that ever was made in mathematics, is

due to John Napier, baron of Merchiston, in Scotland, who published it to the world, in 1614, in a work which he called Mirifici Logarithmorum Canonis Descriptio, and which contained a large table of logarithms, together with their description and uses: but the author reserved his method of constructing them, till the sense of the learned concerning his invention should be known.

17. In the above mentioned work, Napier explains his notion of logarithms by lines described, or generated, by the motion of points, in this manner:-He first conceives a line to be generated by the motion of a point, which passes over equal portions of it, in equal small moments or portions of time. He then considers another line to be generated by the unequal motion of a point, in such a manner, that, in the aforesaid equal portions, or moments of time, there may be described, or cut off from a given line, parts which shall be continually in the same proportion with the respective remainders of that line, which had before been left; then are the several lengths of the first line the logarithms of the corresponding parts of the latter. Which description of them is similar to that which we have already given, viz. that logarithms are a series of quantities, or numbers in arithmetical progression, adapted to another series in geometrical progression.

18. Napier made the first, or whole length of the line, which is diminished in geometrical progression, the radius of a circle; and its logarithm O, or nothing, representing the beginning of the first or arithmetical line. Thus the several proportional remainders of the geometrical line are the natural sines of all arches of a quadrant, decreasing down to 0; while the successive increasing values of the arithmetical line are the corresponding logarithms of those decreasing sines so that, while the natural sines decrease from radius to 0, their logarithms increase from 0 to infinity. Napier made the logarithm of radius to be 0, that he might save the trouble of adding and subtracting it in trigonometrical operations, in which it so frequently occurred; and he made the logarithms of the sines, from the entire quadrant down to 0, to increase, that they might be positive, and so, in his opinion, easier to manage; the sines being of more frequent use than the tangents and secants, of which the whole of the latter, and the half of the former, being greater than radius, would, according to his construction, have their logarithms negative.

19. The description and use of Napier's canon being in the Latin language, they were translated into English by Mr. Edward Wright, the ingenious inventor of what is commonly, though erroneously, called Mercator's sailing. The translation was sent to the author, who revised it, and returned it with his approbation. Mr. Wright, however, dying soon after he received it back, the work, together with the tables, was published in 1616, after his death, by his son Samuel Wright, who dedicated it to the East India Company. It contained also a preface by Henry Briggs, of whom we shall have occasion to speak again presently, on account of the great share he bore in perfecting the logarithms.

20. As Napier's canon contained only the

natural sines for every minute of the quadrant, and their corresponding logarithms, it was attended with some degree of inconvenience, when used as a table of the logarithms of common numbers; because, when a number was proposed which was not exactly the same with some number denoting a natural sine, it was then necessary to find its logarithm by means of an arithmetical calculation, performed according to precepts which the author delivered in his work. This inconvenience, which was in part obviated by certain contrivances of Wright and Briggs, was not the only one; for there was another, which arose from the logarithms being sometimes +, or additive, and sometimes, or negative; and which therefore required the knowledge of algebraic addition and subtraction. This last defect was occasioned partly by making the logarithm of radius 0, and those of the sines to increase; and partly by the compendious manner in which the author had formed the table; making the three columns of sines, cosines, and tangents, to serve also for the other three of cosecants, secants, and cotangents.

21. But this latter inconvenience was well remedied by John Speidell, in his New Logarithms, first published in 1617; which contained all the six columns, and these all of a positive form, by being taken the arithmetical complements of Napier's, that is, they were the remainders left by subtracting each of the latter from 10,000,000. And the former inconvenience was more completely removed by Speidell in a second table, given in the sixth impression of the former work, in the year 1624. This was a table, of Napier's Logarithms for the integers 1, 2, 3, 4, 5, &c., to 1000, together with their differences and arithmetical complements; as also the halves of the said numbers, with their differences and arithmetical complements; which halves were consequently the logarithms of the square roots of the said numbers. Those logarithms are, however, a little varied in their form from Napier's, namely, so as to increase from 1, whose logarithm is 0, instead of decreasing to 1 or radius, whose logarithm Napier made O likewise; that is, Speidell's logarithm of any number n is equal to Napier's logarithm of its reciprocal. So that, in this last table of Speidell's, the logarithm of 1 being 0, the logarithm of 10 is 2302584; the logarithm of 100 is twice as much, or 4605168; and that of 1000, thrice as much, or 6907753. The logarithms contained in this table are now commonly called hyperbolic logarithms; because they serve to express the areas contained between the curve of the hyperbola and its assymptote.

22. The celebrated inventor of the logarithms died in the year 1618; and in 1619 his son Robert Napier published a new edition of the Logarithmorum Canonis Descriptio, to which was now added the promised Logarithmorum Canonis Constructio, and other miscellaneous pieces written by his father and Mr. Briggs. This work was reprinted in France in 1620; also, nearly about this time, different mathematicians abroad published tables of logarithms of the same kind as those of Napier; as Benjamin Ursinus, mathematician to the elector of Brandenburgh,

also the famous Kepler, who was the mathematician to the emperor Ferdinand II. and others. 23. Next to the discovery of logarithms, the most remarkable circumstance connected with their history is that improvement which they received in their form from Henry Briggs, who was, at the time of the publication of Napier's logarithms, professor of geometry in Gresham College, in London, and afterwards Saviliar professor of geometry at Oxford, where he died in the year 1630.

24. On the first publication of Napier's logarithms, Briggs immediately applied himself to the study and improvement of them; and presently saw that it would be of advantage to change the scale; so that the logarithm of 1 being 0, as in Napier's form, the logarithm of 10 might be 1, that of 100 2, of 1000 3, and so on; whereas the logarithms of the same numbers, according to Napier's construction, were 2.302585, 4-605168, 6.907753, &c. This improvement Briggs communicated both to the public in his lectures, and to Napier himself, who afterwards said, that he had also thought of the same thing; as appears from the following extract, translated from the preface to Brigg's Arithmetica Logarithmica: Wonder not,' says he, that these logarithms are different from those which the excellent baron of Marchiston published in his Admirable Canon. For when I explained the doctrine of them to my auditors at Gresham College, in London, I remarked, that it would be much more convenient, the logarithm of the sine total being 0 (as in the Canon Mirificus), if the logarithm of the tenth part of the said radius, namely, of 5° 44′ 21′′, were 100000, &c., and concerning this, I presently wrote to the author; also, as soon as the season of the year, and my public teaching, would permit, I went to Edinburgh, where, being kindly received by him, I staid a whole month. But, when we began to converse about the alteration of them, he said, that he had formerly thought of it, and wished it; but that he chose to publish those that were already done, till such time as his leisure and health would permit him to make others more convenient. And, as to the nature of the change, he thought it more expedient, that O should be the logarithm of 1, and 100000, &c., the logarithm of radius, which I could not but acknowledge was much better; therefore, rejecting those I had before prepared, I proceeded, at his exhortation, to calculate thèse, and the next summer I went to Edinburgh, to show him the principal of them, and should have been glad to do the same the third summer, if it had pleased God to spare him so long.'

25. Thus it appears that Briggs was the inventor of the present scale of logarithms, in which 1 is the logarithm of 10, and 2 that of 100,&c., and that the share which Napier had in them was only advising Briggs to begin at the lowest number 1, and make the logarithms, or artificial numbers (as Napier had also called them), to increase with the natural numbers, instead of decreasing, which made no alteration in the figures that expressed Briggs's logarithms, but only in their affections or signs, changing them from negative to positive; for, according to Briggs's first intention, the logarithms of 001, 01, 1, 1, 10, 100, 1000,

&c., would have been 3, +2, +1, 0, -1, -2, -3, &c.; but, in conformity to the suggestion of Napier, they were made -3, -2, -1, 0, +1, +2, +3, &c., which is a change of no essential importance, as the scale of the system is the same in either case. And the reason why Briggs, after that interview, rejected what he had before done, and began anew, was probably because he had adapted his new logarithms to approximate sines of arcs, instead of the round or integer numbers, and not from their being logarithms of another system, as were those of Napier.

26. About the year 1618 Briggs published the first 1000 logarithms to eight places of figures, besides the index, under the title of Logarithmorum Chilias Prima: and in 1624 he produced his Arithmetica Logarithmica, a stupendous work for so short a time, containing the logarithms of 30,000 natural numbers, to fourteen places of figures besides the index; namely, from 1 to 20,000, and from 90,000 to 100,000, together with the differences of the logarithms; and in both these works the logarithms were calculated according to the system which had been agreed upon between him and the first inventor; that is, they were the same as the system which we commonly employ at the present time.

27. Soon after the publication of the Arithmetica Logarithmica, Adrian Vlacq, or Flack, completed the intermediate seventy chiliads, and republished it at Gouda, in Holland; thus making in the whole, the logarithms of all numbers from 1 to 100,000; but only to ten places of figures. He also added a table of artificial sines, tangents, and secants, to every minute of the quadrant.

28. Briggs himself also lived to complete a table of logarithmic sines and tangents for the 100th part of every degree of the quadrant, to fourteen places of figures, besides the index; together with a table of natural sines for the same parts, to fifteen places, and the tangents and secants of the same to ten places, with the construction of the whole. These tables were printed at Gouda, under the care of Adrian Vlacq, and nearly finished off before 1631. But the death of the author, which happened in 1630, prevented him from completing the application and uses of them. However, the performing of this office he recommended, when dying, to his friend Henry Gellibrand, then professor of astronomy in Gresham College; who added a preface, and the application of the logarithms to plane and spherical trigonometry, &c. The work was published in 1633, under the title of Trigonometria Britannica; and besides the arcs in degrees, and centesms of degrees, it has another column, containing the minutes and seconds, answering to the several centesms in the first column.

29. In the same year Vlacq printed at Gouda his Trigonometria Artificialis; sive Magnus Canon Triangulorum Logarithmicus, ad Decadas Secundorum Scrupulorum constructus. This work contains the logarithmic sines and tangents to ten places of figures, with their differences, for every ten seconds in the quadrant. To them is