Alfo in the Trigonometria of the fame author, printed in 1643, befides the logarithms of numbers from 1 to 1000, to eight places, with their differences, we find both natural and logarithmic fines, tangents and fecants, the former to feven and the latter to eight places; namely, to every 10" of the first 30 minutes, to every 30" from 30 to 10; and the fame for their complements, or backwards through the laft degree of the quadrant; the intermediate 88° being to every minute only.

Mr. Nathaniel Roe, " Paftor of Benacre in Suffolke," alfo reduced the logarithmic tables to a contracted form, in his Tabula Logarithmica, printed at London in 1633. Here we have Briggs's logarithms of numbers from 1 to 100000, to eight places; the fifties placed at top, and from 1 to 50 on the fide; also the first four figures of the logarithms at top, and the other four down the columns. They contain alfo the logarithmic fines and tangents to every 100th part of degrees, to ten places.

Ludovicus Frobenius published at Hamburgh, in 1634, his Clavis Univerfa Trigonometria, containing tables of Briggs's logarithms of numbers from 1 to 2000; and of fines, tangents, and fecants, for every minute; both to feven places.

But the tables of logarithms of common numbers was reduced to its moft convenient form by John Newton, in his Trigonometria Britannica, printed at London in 1658, having availed himself of both the improvements of Wingate and Roe, namely, uniting Wingate's difpofition of the natural numbers with Roe's contracted arrangement of the logarithms, the numbers being all difpofed as in our beft tables at prefent, namely, the units along the top of the page, and the tens down the left-hand fide, alfo the first three figures of each logarithm in the first column, and the remaining five figures in the other columns, the logarithms being to eight places. This work contains alfo the logarithmic fines and tangents, to eight figures befides the index, for every 100th part of a degree, with their differences, and for 1000th parts in the first three degrees. In the preface to this work, Newton takes occafion, as Wingate and Norwood had done before, as well as Briggs himself, to cenfure the unfair practices of fome other publishers of logarithms. He fays, "In the fecond part of this inftitution, thou art prefented with Mr. Gellibrand's Trigonometrie, faithfully tranflated from the Latin copy, that which the author himfelf published under the title of Trigonometria Britannica, and not that which Vlacq the Dutchman ftiles Triganometria Artificialis, from whofe corrupt and imperfect copy that seems to be tranflated, which is amongst us generally known by the name of Gellibrands Trigonometry, but those who either knew him, or have perufed his writings, can teftifie that he was no admirer of the old fexagenary way of working, nay, that he did preferre the decimal way before it, as he hath abundantly teftified in all the examples of this his Trigonometry, which differs from that other which Vlacq hath published, and that which hath hitherto borne his name in English, as in the form; (o likewife in the matter of it; for in the two laft-mentioned editions, there is fome

thing left out in the fecond chapter of plain triangles, the third chapter wholly omitted, and a part of the third in the fpherical, but in this edition nothing, fomething we have added to both, by way of explanation and demonftration."

In 1670, John Caramuel published his Mathefis Nova, in which are contained 1000 logarithms both of Napier's and Briggs's form, as alfo 1000 of what he calls the Perfect Logarithms, namely the fame as thofe which Briggs first thought of, which differ from the last only in this, that the one increases while the other decreafes, the radix, or logarithm of the ratio of 10 to 1, being the fame in both.

The books of logarithms have fince become very numerous, but the logarithms are moftly of that fort invented by Briggs, and which are now in common ufe. Of these the most noted for their accuracy or usefulness, befides the works above-mentioned, are Vlacq's fmall volume of tables, particularly that edition printed at Lyons in 1670; also tables printed at the fame place in 1760; but most efpecially the tables of Sherwin and Gardiner. Of thefe, Sherwin's Mathematical Tables in 8vo, form the most compleat collection of any, containing, befides the logarithms of all numbers to 101000, the fines, tangents, fecants, and verfed fines, both natural and logarithmic, to every minute of the quadrant. The firft edition was in 1706; but the third edition, in 1742, which was revifed by Gardiner, is eftcemed the most correct of any as to the laft or fifth edition, in 1771, it is fo erroneously printed, that no dependence can be placed in it, and it is the moft inaccurate book of tables I ever knew; I have a lift of feveral thoufand errors which I have corrected in it.

Gardiner alfo printed at London, in 1742, a quarto volume of "Tables of Logarithms, for all numbers from 1 to 102100, and for the fines and tangents to every ten feconds of each degree in the quadrant; as alfo, for the fines of the first 72 minutes to every fingle fecond with other ufeful and neceflary tables;" namely, a table of Logistical Logarithms, and three fmaller tables to be used for finding the logarithms of numbers to twenty places of figures. Of thefe tables of Gardiner, only a fmall number was printed, and that by fubfcription; and they are now in the highest eftimation of any logarithms for their accuracy and usefulness.

An edition of Gardiner's collection was alfo elegantly printed at Avignon in France, in 1770, with fome additions, namely, the fines and tangents for every fingle fecond in the first four degrees, and a fmall table of hyperbolic logarithms copied from a treatife on Fluxions by the late ingenious Mr. Thomas Simpfon: but this is not quite fo correct as Gardiner's own edition. The tables in all these books are to seven places of figures.

"The logarithmic canon ferves to find readily the logarithm of any affigned number; and we are told by Dr. Wallis, in the fecond volume of his Mathematical Works, that an antilogarithmic canon, or one to find as readily the number correfponding to every logarithm, was begun he thinks by Mr. Harriot the algebraift (who died in 1621) and completed by Mr. Walter Warner, the editor of Harriot's works,

before 1640; which ingenious performance it feems was loft, for want of encouragement to publish it."

"A fmall fpecimen of fuch numbers was published in the Philofophical Transactions, for the year 1714, by Mr. Long of Oxford; but it was not till 1742 that a complete antilogarithmic canon was publifhed, by Mr. James Dodfon, wherein he has computed the numbers correfponding to every logarithm from 1 to 100000, to 11 places of figures."

Since the preceding account was written, and whilft it was in the prefs, there has been printed at Paris, "Tables Portatives de Logarithmes, publiées a Londres par Gardiner," &c. This work is most beautifully printed in a neat portable 8vo volume, and contains all the tables in Gardiner's 4to. volume, with fome additions and improvements. But with what degree of accuracy remains yet to be determined. And on this, as well as feveral other occafions, it is but justice to remark the extraordinary fpirit and elegance with which the learned men and the artifans of the French nation undertake and execute works of merit.

THE

CONSTRUCTION

O F

LOGARITHM S,

HAVING

&c.

[AVING defcribed the feveral forts of logarithms, their rife and invention, their nature and properties, and given fome account of the principal early cultivators of them, with the chief collections that have been published of fuch tables; I proceed now to deliver a more particular account of the ideas and methods employed by each author, and the peculiar modes of conftruction which they made ufe of. And firft of the great inventor himself, lord Napier.

Napier's Construction of Logarithms.

The inventor of logarithms did not adapt them to the feries of natural numbers 1, 2, 3, 4, 5, &c, as it was not his principal idea to extend them to all arithmetical operations whatever; but he confined his labours to that circumftance which firft fuggefted the neceffity of the invention, and adapted his logarithms to the approximate numbers expreffing the natural fines of every minute in the quadrant, as they had been fet down by former writers on trigonometry.

The fame reftricted idea was pursued through his method of conftructing the logarithms. As the lines of the fines of all arcs, are parts of the radius, or fine of the quadrant, therefore called the finus totus or whole fine, he conceived the line of the radius to be defcribed, or run over, by a point moving along it in fuch manner, that in equal portions of time it generated, or cut off, parts in a decreafing geometrical progreffion, leaving the feveral remainders, or fines, in geometrical progreffion alfo; whilft another point, in an indefinite line, described equal parts of it in the fame equal portions of time; fo that the refpective fums of these, or the whole line generated, were always the arithmeticals or

I

2

Al

logarithms of those fines. Thus, az is the given radius upon Sines. Log. which all the fines are to be taken, and A&c the indefinite a line containing the logarithms; these lines being each generated by the motion of a point, beginning at A, a. Now at the end of the 1ft, 2d, 3d, &c, moments, or equal small portions of time, let the moving points be found at the places marked 1, 2, 3, &c; then za, z1, z2, z3, &c, will be the feries of natural fines, and Ao (or o), A1, A2, A3, &c, will be their logarithms; fuppofing the point which generates az to move every where with a velocity decreafing in proportion to its diftance from z, namely, its velocity in the points o, 1, 2, 3, &c, to be refpectively as the diftances zo, Z1, Z2, Z3, &c, whilft the velocity of the point generating the logarithmic line A&c, remains constantly the same as at firft in the point A or o.

7

&C

&c

Hitherto the author had not fully limited his fyftem or scale of logarithms, having only fuppofed one condition or limitation, namely, that the logarithm of the radius az fhould be o. But two independant conditions, no matter what they are, were neceffary to limit the scale or fyftem of logarithms. It did not occur to him, that it was proper to form the other limit by affixing fome particular value to an affigned number, or part of the radius: but as another condition was neceffary, he affumed this for it, namely, that the two generating points fhould begin to move at a, A with equal velocities; or that the increments ai, Ai, described in the first moments, should be equal; as he thought this circumftance would be attended with fome little ease in the computation. And this is the reafon that, in his table, the natural fines and their logarithms, at the compleat quadrant, have equal differences; and this is alfo the reafon why his fcale of logarithms happens accidentally to agree with what have fince been called the hyperbolic logarithms, which have numerical differences equal to thofe of their natural numbers at the beginning; except only that these latter increase with the natural numbers, and his on the contrary decrease; the logarithms of the ratio of 10 to 1 being the fame in both, namely 2.30258509.

And here by the way it may be obferved, that Napier's manner of conceiving the generation of the lines of the natural numbers and their logarithms, by the motion of points, is very fimilar to the manner in which Newton afterwards confidered the generation of magnitudes in his doctrine of fluxions; and it is alfo remarkable that, in art. 2. of the Habitudines Logarithmorum & fuorum naturalium numerorum invicem in the appendix to the Conftructio Logarithmorum, Napier fpeaks of the velocities of the increments or decrements of the logarithms, in the fame way as Newton does, namely of his fluxions, where he fhews that thofe velocities, or fluxions, are inverfely as the fines or natural numbers of the logarithms; which is a neceffary confequence of the nature of the generation of thofe lines as defcribed above; with this alteration however, that now the radius az must be confidered as generated by an equable motion of the point, and the indefinite line A &c by a motion increafing in the fame ratio as the

other before decreafed; which is a fuppofition that Napier must have had in view when he stated that relation of the fluxions.

Having thus limited his fyftem, Napier proceeds, in the pofthumous work of 1619, to explain his conftruction of the logarithmic canon; and this he effects in various ways, but chiefly by generating, in a very eafy manner, a feries of proportional numbers and their arithmeticals, or logarithms; and then finding, by proportion, the logarithms to the natural fines, from thofe of the nearest numbers among the original proportionals.

After defcribing the neceffary cautions he made ufe of to preserve a fufficient degree of accuracy, in fo long and complex a procefs of calculation; fuch as annexing feveral ciphers, as decimals feparated by a point to his primitive numbers, and rejecting the decimals thence refulting after the operations were compleated, fetting the numbers down to the nearest unit in the laft figure; and teaching the arithmetical proceffes of adding, fubtracting, multiplying, and dividing the limits between which certain unknown numbers muft lie, fo as to obtain the limits between which the refults muft alfo fall; I fay, after defcribing fuch particulars, in order to clear and fmooth the way, he enters on the great field of calculation itself. Beginning at radius 10000000, he firft conftructs feveral defcending geometrical feries, but of fuch a nature that they are all quickly formed by an eafy continual fubtraction, and a divifion by 2, or by 10, or 100, &c, which is done by only removing the decimal point fo many places towards the left hand, as there are ciphers in the divifor. He conftructs three tables of fuch feries: The firft of these confifts of 100 numbers, in the proportion of radius to radius minus 1, or of 10000000 to 9999999; all which are found by only fubtracting from each its 10000000th part, which part is alfo found by only removing each figure 7 places lower: the last of these 100 proportionals is found to be 9999900*0004950.

I 10000000.0000000

2

4

2d Table. 0000000.000000

9999999.0000000 9999900* ̄ ̄ ̄ ̄ ̄D 9999998.0000001 9999800 C01000 9999997.0000003 9999700003000 &c till the 100th &c to the 50th term term, which will be 9995001.222927 100 9999900.0004950

&c

The 2nd table contains 50 No. First Table. numbers, which are in the continual proportion of the first to the laft in the first table, namely, of 10000000.0000000 to 9999900.0004950, or nearly the proportion of 100000 to 99999; thefe therefore are found by only removing the figures of each number 5 places lower, and fubtracting them from the fame number: the laft of thefe he finds to be 9995001.222927. And a fpecimen of thefe two tables is here

annexed.

The 3d table confifts of 69 columns, and each column of twentyone numbers or terms, which terms, in every column, are in the continual proportion of 10000 to 9995, that is, nearly as the first is to the laft in the 2d table; and as 10000 exceeds 9995 by the 2000th part, the terms in every column will be conftructed by dividing each upper number by 2, removing the figures of the quotient 3 places lower, and then fubtracting them; and in this way it is proper to conftru&t only the first column of 21 numbers, the laft of which will be