GENERAL DEFINITION.

(1.)*LOGARITHMS. n. f. logarithme, Fr. xoves and agios-Logarithms, which are the indexes of the ratios of numbers one to another, were first invented by Napier lord Merchifton, a Scottish baron, and afterwards completed by Mr Briggs, Savilian profeffor at Oxford. They are a feries of artificial numbers, contrived for the expedition of calculation, and proceeding in an arithmetical proportion, as the numbers they answer to do in a geometrical one. The addition and fubtraction of logarithms answers to the multiplication and divifion of the numbers they correfpond with; and this faves an infinite deal of trouble. In like manner will the extraction of roots be performed, by bifecting the logarithms of any numbers for the fquare root, and trifecting them for the cube, and fo on. Harris.

THE doctrine of logarithms being of great importance in the fcience 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 feries of numbers, the one conftituting an arithmetical progreffion, and the other a geometrical progreffion, 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 ftand over each other in fuch a manner, that o in the arithmetical series correfponds to unity in the geometrical series; then we fhall readily perceive, from induction, that the two feries, fo arranged, poffefs the following proporties:

(3) I. Let the fum of any two terms of the arithmetical feries be taken; and alfo the product of the correfponding terms of the geometrical feries; then, below that term of the arithmetical feries, which is equal to the fum, will be found a term of the geometrical feries, equal to the product.

Thus, if the terms of the arithmetical feries be 3 and 5, thofe of the geometrical feries will be 8 and 32. Now 3+5=8; and 8X32256; and, by infpecting the two feries, we find that the term 256 in the geometrical series stands below 8 of the arithmetical feries.

(4.) II. Let the difference of any two terms of the arithmetical series be taken, and alfo the quo tient of the corresponding terms of the geometri cal feries; then, below that term of the arithmetical feries, which is equal to the difference, will be found a term of the geometrical feries, equal to the quotient.

Thus, if the terms of the arithmetical feries be 5 and 8, and therefore those of the geometrical feries 32 and 256; we shall have 8-5-3, and 256-32-8; and we find, by infpecting the fe ries, that 8 of the geometrical feries ftands below 3 of the arithmetical feries. This laft property is evidently nothing else than the converfe of the former.

(5) In the preceding geometrical feries the common ratio is 2, but it may be any other num.

ber whatever, whole or fractional. Thus the fame properties will be found to hold true of thefe feries:

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 feries is 3. They also hold true in the following: Arith. prog. o, I, 2, 3, 4, 5, &c. Geom. prog. 1, in rip oiu TITT, &C. where the common ratio is.

(6.) To demonftrate that the two foregoing properties muft neceffarily be true in every cafe, it is only neceffary to write down a geometrical feries, according to the algebraic method of notation, thus, or 1, r', r2, r3, r^, r3, 1o, 17, &c. where r denotes the ratio of the feries, and it prefently appears, that the arithmetical series is fupplied by the numeral exponents of the terms. Hence it follows, that the properties, which we have afcribed to any geometrical, and its corres ponding arithmetical feries, are no other than two well known propofitions in algebra; namely, that the fum of the exponents of any two powers of their product; and that the difference of of an algebraic quantity is equal to the exponent their exponents is equal to the exponent of their quotient.

(7.) When the terms of an arithmetical progreflion are adapted to those of a geometrical progreflion, as in the three examples given in § 2 and 5, the terms of the arithmetical feries are called the Logarithms of the corresponding terms of the geometrical series.

Thus, in the first example, o, 1, 2, 3, &c. are the logarithms of 1, 2, 4, 8, 16, &c. refpectively.

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, o, 1, 2, 3, &c. are the logarithms of 1, 3, 5, ris, &c.

By applying now the properties of these feries, which were demonftrated in § 6, to logarithms, and their correfponding numbers, we may define logarithms to be a series of numbers in arithmeti cal progreffion, fo adapted to another feries of numbers in geometrical progreflion, that the fums and differences of the former, correspond to and fhew the products and quotients of the latter.

(8.) From this definition of logarithms it ap pears, that there may be an infinity of different fyftems according as one or another geometrical feries is adapted to the arithmetical feries, 0, 1, 2, 3, &c. It may however be readily fuppofed that fame fyftems are better fuited to general caiculation 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 whichmay have their logarithms expreffed by integers being as in the following table: Logarithms o, I, 2,

31 4, &c. Numbers I, 10, 100, 1000, 10,000, &c. (9.). With refpect to the numbers between 1 and ro, 10 and 100, &c. and their correfponding logarithms, they may be understood to be supplied by interpolation, thus: Conceive a great number of geometrical proportionals to be inferted be tween the natural numbers 1 and 10, which are

the

the extremes; alfo an equal number of arithmetical proportionals between their logarithms o and 1: Then, if the number of geometrical proportionals be fufficiently great, fome one or other of them will be fufficiently near to each of the natural numbers 2, 3, 4, 5, &c. to 9, fo as to admit of the one being taken for the other, without any fenfible error. There will alfo be a correfponding logarithm to each, which, as it will be lefs than mity, may be moft conveniently expreffed by a decimal fraction.

(10.) Let us fuppofe the number of geometrical proportionals between 1 and 10, and alfo the number of arithmetical proportionals between o and 1, to be 9999, and therefore the number of terms, including 1 and 10, 10001. Then the 3011th term of the geometrical feries will be 1'9999, or 2 nearly, and the correfponding term of the arithmetical feries, 3010: Therefore the logarithm of 2 is 3010, nearly. Again, the 4772d term of the geometrical feries will be 2'9999, or 3 nearly; and the correfponding term of the arithmetical feries 4771; therefore the logarithm of 3 is 4771 nearly; and fo on with refpect to other numbers.

(11.) If we fuppofe the series of natural numbers I, 2, 3, 4, 5, &c. to be arranged in a table, fo that each number may stand oppofite to its correfponding logarithm; it is evident from the properties which we have fhewn to belong to logarithms, that, by means of fuch a table, the arithmetical operations of multiplication, divifion, involution, and evolution, may be performed with great facility.

(12.) For, fince the fum 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 oppofite to that logarithm which is the fum of the logarithms of the numbers.

Again, because the difference of the logarithms of two numbers is equal to the logarithm of the quotient arifing from the divifion of the one number by the other, 4; that quotient will be found in the table, oppofite to the logarithm which is excefs of the logarithm of the dividend, above that of the divifor.

SECT. II. HISTORY of LOGARITHMS.

(15.) The properties of a geometrical feries, which conftitute 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 ufe of them in his work entitled Arenarius, or Treatise on the number of fands. The fame properties are alfo mentioned in the writings of STIFELIUS, a German mathematician, who lived about the middle of the 16th century. It does not, however, appear, that any perfon perceived all the advantages which might be derived from these properties, till about the. beginning of the 17th century, when their utility was rendered evident by the happy invention of logarithms.

(16.) This discovery, certainly one of the moft 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 Defcriptio, and which contained a large table of logarithms, together with their defcription and ufes: but the author referved his method of conftructing them, till the fenfe of the learned concerning his invention fhould be known.

(17.) In the abovementioned work, Napier explains his notion of logarithms, by lines defcribed, or generated, by the motion of points, in this manner. He firft conceives a line to be generated by the motion of a point, which paffes over equal portions of it, in equal small moments or portions of time. He then confiders another line to be generated by the unequal motion of a point, in fuch a manner, that, in the aforefaid equal portions, or moments of time, there may be defcribed, or cut off from a given line, parts which fhall be continually in the fame proportion with the respective remainders of that line, which had before been left; then are the feveral lengths of the first line the logarithms of the correfponding parts of the latter. Which defcription of them is fimilar to that which we have already given, viz. that logarithms are a series of quantities, or numbers in arithmetical progreffion, adapted to another series in geometrical progreffion.

(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 correfponding logarithms, the operations of multiplication and divifion may be reduced to the more fimple operations of addition and fubtraction; and the operations of involution and evolution, to thofe of multiplication and divifion.

VOL. XIII. PART I.

gents,

gents and fecants, of which the whole of the latter, and the half of the former, being greater than radius, would, according to his conftruction, have their logarithms negative.

(19.) The defcription and use of Napier's canon being in the Latin language, they were tranflated into English by Mr EDWARD WRIGHT, the ingenious inventor of what is commonly, though erroneously, called MERCATOR'S SAILING. The tranflation was fent to the author, who revised it, and returned it with his approbation. Mr Wright, however, dying foon after he received it back, the work, together with the tables, was published in 1616, after his death, by his fon Samuel Wright, who dedicated it to the Eaft India Company. It contained alfo a preface by HENRY BRIGGS, of whom we shall have occafion to speak again prefently, on account of the great fhare he bore in perfecting the logarithms.

(20.) As Napier's canon contained only the natura! fines for every minute of the quadrant, and their correfponding logarithms, it was attended with fome degree of inconvenience, when used as a table of the logarithms of common numbers; because when a number was propofed, which was not exactly the fame with fome number denoting a natural fine, it was then neceffary 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 arofe from the loga. rithms being fometimes +, or additive, and fometimes, or negative; and which therefore required the knowledge of algebraic addition and fubtraction. This laft defect was occafioned partly by making the logarithm of radius o, and thofe of the fines to increase; and partly by the compendious manner in which the author had formed the table; making the three columns of fines, cofines, and tangents, to ferve alfo for the other three of cofecants, fecants, and cotangents.

(21.) But this latter inconvenience was well remedied by JOHN SPEIDELL, in his New Logarithms, firft published in 1617; which contained all the fix columns, and thefe all of a pofitive form, by being taken the arithmetical complements of Napier's, that is, they were the remainders left by fubtracting each of the latter from 10,000,000. And the former inconvenience was more completely removed by Speidell in a fecond table, given in the fixth impreffion 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 alfo the halves of the faid numbers, with their differences and arithmetical complements; which halves were confequentby the logarithms of the fquare roots of the faid numbers. Thofe logarithms are, however, a little varied in their form from Napier's, namely, fo as to increase from 1, whofe logarithm is o, inftead of decreafing to 1 or radius, whofe logarithm Napier made o likewife; that is, Speidell's logarithm of any number n is equal to Napier's loga

rithm of its reciprocal So that, in this last ta

ble of Speidell's, the logarithm of 1 being o, 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 ferve to exprefs the areas contained between the curve of the hyperbola and its affymptote.

(22.) The celebrated inventor of the logarithms died in the year 1618; and in 1619, his fon Robert Napier published a new edition of the Logarithmorum Canonis Defcriptio, to which was now added the promised Logarithmorum Canonis Conftructio, and other mifcellaneous pieces written by his father and Mr Briggs. This work was reprinted in France, in 1620; alfo, nearly about this time, different mathematicians abroad published tables of logarithms of the fame kind as thofe of Napier; as BENJAMIN URSINUS, mathematician to the Elector of Brandenburg, also the famous KEPLER, who was the mathematician to the emperor Ferdinand II. and others.

(23.) Next to the difcovery of logarithms, the moft remarkable circumftance connected with their hiftory, is that improvement which they received in their form from HENRY BRIGGS, who was, at the time of the publication of Napier's logarithms, profeffor of geometry in Gresham College, in London, and afterwards Savilian profeffor of geometry at Oxford, where he died in the year 1630.

(24.) On the first publication of Napier's loga rithms, Briggs immediately applied himself to the ftudy and improvement of them; and prefently faw that it would be of advantage to change the fcale; fo that the logarithm of being o, as in Napier's form, the logarithm of 10 might be I, that of 100 2, of 1000 3, and fo on; whereas the logarithms of the fame numbers, according to Napier's conftruction, were 2'302585, 4*605168, 6907753, &c. This improvement Briggs communicated both to the public in his lectures, and to Napier himself, who afterwards faid, that he had alfo thought of the fame thing; as appears from the following extract tranflated from the preface to Briggs's Arithmetica Logarithmica. "Wonder not," fays he, "that thefe logarithms are different from thofe which the excellent baron of Marchifton publifhed in his Admirable Canon. For when I explained the doctrine of them, to my auditors at Grefham College in London, I remark ed, that it would be much more convenient, the logarithm of the fine total being o(as in the Canon Mirificus), if the logarithm of the roth part of the faid radius, namely of 5° 44' 21", were 100000, &c. and concerning this, I prefently wrote to the author; alfo, as foon as the feafon of the year, and my public teaching, would permit, I went to Edinburgh, where, being kindly received by him, I ftaid a whole month. But when we began to converse about the alteration of them, he faid, that he had formerly thought of it, and wished it; but that he chofe to publifh thofe that were already done, till fuch 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 fhould be the logarithm of 1, and 100000, &c. the logarithm of radius, which I could not but acknowledge was much better,

Therefore,

Therefore, rejecting thofe I had before prepared, I proceeded, at his exhortation, to calculate thefe; and the next fummer I went to Edinburgh, to fhew him the principal of them, and fhould have been glad to do the fame the third fummer, if it had pleafed God to fpare him fo 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 fhare which Napier had in them was only advifing Briggs to begin at the lowest number 1, and make the logarithms, or artificial numbers, (as Napier had also called them), to increafe with the natural numbers, instead of decreafing, which made no alteration in the figures that expreffed Briggs's logarithms, but only in their affections or figns, changing them from negative to pofitive; for, according to Briggs's firft intention, the logarithms of oo1, 01, 1, 1, 10, 100, 1000, &c. would have been +3, +2, +1, 0, -1, -2, -3, &c. but, in conformity to the fuggeftion of Napier, they were made -3, -2, −1, 0, +1, +2, +3, &c. which is a change of no effential importance, as the fcale of the fyftem is the fame in either cafe. And the reafon 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 fines of arcs, instead of the round or integer numbers, and not from their being logarithms of another fyftem, as were thofe of Napier. (26.) About the year 1618, Briggs publifhed the first thousand logarithms to eight places of figures, befides the index, under the title of Logarithmorum Chilias Prima. And, in 1624, he produced his Arithmetica Logarithmica, a ftupendous work for fo fhort a time, containing the logarithms of 30,0co natural numbers, to 14 places of figures befides 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 fyftem which had been agreed upon between him and the first inventor; that is, they were the fame as the fyftem which we commonly employ at the prefent time.

(27-) Soon after the publication of the Arithmetica Logarithmica, ADRIAN VLACQ, or FLACK, completed the intermediate 70 chiliads, and republished it at Gouda, in Holland; thus making, in the whole, the logarithms of all numbers from I to 100,000; but only to ten places of figures. He alfo added a table of artificial fines, tangents, and fecants, to every minute of the quadrant.

(28.) BRIGGS himself alfo lived to complete a table of logarithmic fines and tangents for the 100th part of every degree of the quadrant, to 14 places of figures, befides the index; together with a table of natural fines for the fame parts, to 15 places, and the tangents and fecants for the fame to 10 places, with the conftruction of the whole. These tables were printed at Gouda, under the care of Adrian Viacq, and nearly finished off before 1631. But the death of the author, which happened in 1630, prevented him from completing the application and ufes of them. However, the performing of this office he recommended, when dying, to his friend HENRY GELLIBRAND, then

professor of aftronomy in Gresham college; who added a preface, and the application of the lega rithms to plane and spherical trigonometry, &e. The work was published in 1633, under the title of Trigonometria Britannica; and befides the arcs in degrees, and centefms of degrees, it has another column, containing the minutes and feconds, anfwering to the feveral centesms in the firft column.

(29.) In the fame year, Vlacq printed at Gouda his Trigonometria Artificialis; five Magnus Canon Triangulorum Logarithmicus, ad Decadas Secundorum Scrupulorum conftruclus. This work contains the logarithmic fines and tangents to ro places of figures with their differences, for every re feconds in the quadrant. To them is alfo added Briggs's table of the firft 20,000 logarithms; but carried only to 10 places of figures, befides the index, with their differences. The whole is preceded by a description of the tables, and the application of them to plane and ipherical trigonometry, chiefly extracted from Briggs's Trigonometria Britannica, mentioned above.

(30.) GELLIBRAND published also, in 1635, An Inftitution Trigonometrical, containing the logarithms of the first 10,000 numbers, with the natural fines, tangents, and fecants, and the logarithmic fines and tangents for degrees and minutes, all to places of figures, befides the index; as alfo other tables proper for navigation, with the ufes of the whole.

(31.) Having now given some account of fuch works on logarithms, as feem molt connected with their firft difcovery and fubfequent improvement, we shall pass over many others, some of which, however, have been held in high repute, both for their accuracy, and the extent to which the tables have been carried. As, however, even the arrangement of the logarithms in the tables has received confiderable improvements fince the days of Napier, it may be proper to mention, that they were firft reduced to the moft convenient form by JOHN NEWTON, in his Trigonometria Britannica, publifhed at London in 1658.

(32.) Among the tables of logarithms which have been published of late years in this country, there are two works most deservedly in repute, both for accuracy and convenience of arrange ment: Thefe are, Dr HUTTON'S Mathematical Tables, containing Common, Hyperbolic and Logistic Logarithms, &c. and TAYLOR'S Tables of Loga rithms of all numbers, from 1 to 101,000; and of the Sines and Tangents to every fecond of the quadrant. Several very accurate and well arranged collections of tables of logarithms have also been lately printed in France; one, which deferves to be particu larly mentioned, is CALLET's ftereotype edition of Tables Portatives de Logarithms. These contain the logarithms of numbers from 1 to 108,000, and the logarithmic fines and tangents for every fecond, in the first 5 degrees, and for every 10 fe conds of the remaining degrees of the quadrant, and alfo for every 10000th part of the arc, according to the new centefimal divifion of the quadrant. The logarithms are to 7 decimal places.

(33) But a more extenfive collection of logarithmic tables than any we have yet mentioned, was begun in France, in 1794, under the direction of C. PRONY, who engaged, "not only to compose

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