In 1578 William Bourne published Inventions and Devices. There are one hundred and thirteen subjects treated of, many of them highly interesting, as they contain the crude germ of useful inventions. The twenty-first device is a close approach to Massey’s self-registering log, which was found so useful two hundred and sixty years later. The credit of the device is ascribed to Humfray Cole; the probable date is 1570. The proposal was to have a “little small close boat” with a wheel, or wheels, and an axletree, to turn clock-work in the little boat, with dials and pointers to indicate respectively fathoms, leagues, scores of leagues, and hundreds of leagues. If a small screw rotator had been used instead of a wheel, this might have been a great success. It was only a suggestion, perhaps untried; and in common with seamen and writers about that time the author allows only 5000 feet to a mile. Edward Wright’s Certain Errors in Navigation detected and corrected (1610) gives much new and useful information, but the nearest allusion to the ship’s speed is in the part translated from the Spanish of Roderigo Samorano, under the head of finding the ship’s place on the chart, called the “point of imagination.” “This point doth presuppose the knowledge of two things: to wit, the rhumb by which we have sailed, and that is known by the compass, and the leagues which we have run; and this hath no certainty, but is a little more or less than a good mariner according to his imagination supposeth that he hath sailed; whereof the said point took its name.” In 1624 an edition of Gunter by Edward Weaver, after much valuable geometric information, proposes at chap, vi., in a long rambling manner, that an account should be kept of the ship’s way. “The way that a ship maketh may be known to an old seaman by experience, by others it may be found,” as he recommends, with the log-line or by known marks on the ship’s side, bearing the proportion to a league or mile, that a certain number of seconds do to an hour. So far good; but he reckons a mile as 5866 feet (214 too little), and states that seamen count in paces of 5 feet each, and 1000 to a mile, i.e., only 5000 feet. He also proposes to divide the degrees into one hundred parts, each to be called “centesmes.” The whole subject is treated as a new thing. It is stated by Purchas (1625) that Christopher Columbus (1492) deceived his crew with respect to the distance sailed from home, and that “even the pilots did not know how far they had gone” as they glided so smoothly with a continuous fair wind. Had any kind of log been hove, the ship’s speed would have been publicly known. Mr Burnaby {Ancient Geography, p. 554) states that “no ancient writer has preserved any account of the mode in which ancient navigators computed distance.” Following such an authority and the quotations above, we may safely agree with the statement of Purchas that it was first used in 1607. Also we know that it did not become general till many years after. In one of our best works on navigation, printed in 1843, the log is inaccurately described.

If we are surprised that so many centuries passed, and that long voyages were made, after the discovery of the compass, without any means of measuring the distance sailed, we may be almost as much so at the diversity of opinion which prevailed among seamen with regard to the length of the log-line and the length of a mile. At the present day the principle upon which this log is arranged is easily understood. The mean degree of the meridian (see vol. x. p. 198) is assumed to be 69Ό9 statute miles, which gives 6080 feet to the mean nautical mile,—an estimate sufficiently accurate for navigating upon any part of the sphere. The distances upon the log-line being marked by pieces of line placed between the strands and carrying the requisite number of knots, this has given the name of knot to the nautical mile. The line is marked to knots and half knots (a single knot) only; the intermediate fractions are estimated. Two measurements are now in common use; that in the British navy is 47 feet 3 inches of line for each knot made per hour, which corresponds with a twenty-eight second glass—thus (28 × 6080) ÷3600 = 47,288 feet; in the merchant service a knot is 50 feet 7 inches, which is the correct proportion to a mile with the half minute glass. When a ship is going more than five or six knots, a short glass is used, fourteen or fifteen seconds, then the indications by the line are doubled. The shorter measure was probably chosen in consequence of the custom in vogue till about 1833 of marking the run on the log-board, or book, in knots and fathoms (or sea furlongs); the fractions are now invariably entered as tenths. The whole length of line is 60 to 80 fathoms, according to the speed anticipated; 10 to 20 fathoms of which is allowed as stray line, that the log-ship may be in a fair position, before the rag of bunting called the turn mark passes the hand. The line should be stretched and well wet before it is measured, and should be remeasured every day at sea. The inner end of the line is made fast to a light reel upon which it is wound.

The “log-ship” (fig. 1) is a piece of wood about ⅜ inch thick and the fourth part of a circle, having a radius of 5 or 6 inches, weighted with lead round the curve in order to keep it upright in the water, but not to sink it. Two holes are bored, about 1⅛ inches from the lower angles; through one a short piece of line is passed and knotted; the other end of the line has a bone or hard peg spliced to it, which . is inserted in the other hole, thus forming a span t^ 1S a ^ ac hed to the log-line, and hangs square.

When the log is used,> man holds the reel over his head, the officer places the peg in the log-ship, and throws it Well clear of the wake, then allows it to run the “stray line” off without assistance, steadying it just before the turn mark comes to hand; as the mark passes he calls to his assistant with the glass to “turn.” As the sand runs pay out freely till the word “stop” is expected, then bring the line into a state of tension similar to what it was in when the turn mark passed. At the word “stop” nip the line instantly, count the nearest knots, and estimate the tenths. When the line is stopped the strain should cause the peg to draw from the log-ship, and it can easily be hauled in. In ships of war it is hove every hour. The value of the operation depends, of course, entirely upon the care bestowed.

Ground-Log.— In large rivers, such as Rio de la Plata, where a strong current runs, and shoals are found out of the sight of land, a lead of four or five pounds weight is used instead of the log-ship; the lead rests on the bottom, the line and sand-glass being used in a manner similar to that above described. This is called the ground-log, and indicates the speed at which the ship is passing over the ground, irrespective of currents or tides; it will show also the lateral effect of current as it is hauled in; this is the only log which can do so.

The sand-glasses are very primitive contrivances for measuring the requisite number of seconds; they are much affected by damp and change of temperature, and no reliance can be placed on their accuracy. In 1868 a timepiece sounding a gong at the required intervals was devised by the late Admiral Sir Walter Tarleton, and was tried on board some of Her Majesty’s ships, but failed after a short time from damp or other causes. The writer of this article was then attempting to produce a log-gong, but abandoned it on being told that they could be obtained below his estimated cost.

Screw Logs.— In 1725 Henry de Saumarez described a machine which was to supersede the ordinary log. This was on the principle of the screw, having vanes which caused it to revolve and communicate a rotary motion to a piece of rope; this most probably went inboard to clockwork; hence the failure. Mr Smeatonmade many experiments about 1751; he found the results very irregular, especially at high velocities, just as the writer of this article did with one of Massey’s flies and a line or wire attached to a spindle, supported by large friction rollers inboard; both experiments were dead failures on account of the friction. In 1773 two screw logs were tried on board H.M. ship “Racehorse” during a short voyage to the Polar Sea; each was made to communicate motion to a counter inboard. In 1792 Mr Gower took out a patent for a screw log. None of these experiments were sufficiently successful to gain the confidence of seamen.

We see that the principle was not new in 1834 when Mr Massey patented a screw log, which has been so generally adopted that it deserves special notice and description. Though Massey took out other patents, and others have followed with modifications, the principle of all is the same, and likely to remain in use with the “common log” for many years to come. Massey fitted his log to the sternpost of a few vessels, a vertical spindle conveying the rotations of the fly to a register in the cabin above, but it did not answer.

The log of 1836 which came into general use is represented in fig. 2. It consists of two parts united by 2 or 3 feet of rope. The “fly” consists of a hollow copper cylinder about 9 or 10 inches long with four fins or blades placed at a given angle, causing it to rotate once in a certain distance. The rope is attached to the fly and to a spindle which freely revolves in a brass box; an endless screw acting upon a system of wheel-work records the fractions of a mile on one dial, units upon a second, and tens up to one hundred on the third, on the same principle with the index of a gas meter.

The last patent was for the “frictionless log” shown in fig. 3, which is similar to the former except that, by dispensing with the piece of rope and part of the heavy box, it is much more compact and less liable to foul,—an accident to which all logs when towed after a ship are very liable. Walker’s harpoon log is very similar to the last of Massey’s, but has a plate at the back in the shape of a harpoon to prevent the upper part from revolving. This log is now supplied to Her Majesty’s ships. The fins or blades which cause the rotation in each of the logs above described are flat pieces of brass (not portions of a screw) soldered to a cylinder, which is hollow in order to diminish the tendency to sink when going slowly; but if the log be left overboard when the ship stops, the tow-line will allow it to sink about 100 feet, the pressure of water will then fill it, and there is no means provided for getting the water out. Screw logs will also at low speeds hang obliquely and be useless. Mr Friend tried a log with paddles protruding from a brass box instead of using a screw; but the plan was not adopted. However accurate the registering logs may be, an hourly log cannot be dispensed with, unless the ship be on one course during the whole twenty-four hours, or her speed be uniform; even then the old log and line should not be neglected. Both Massey and Walker are now trying logs the rotators of which are towed, while the dials for registering are on the ship’s stem.

Pressure Bog.— In 1849 the Rev. E. L. Berthon patented a log (fig. 4), indicating the speed of the ship by means of the pressure of water due to the velocity acting upon a tube about f of an inch diameter in the clear, closed at the end, protruding some 8 inches below the ship’s bottom, with an aperture of about an ∣ of an inch in diameter in the front side, near the closed end. A vane was used to turn the aperture in the direction of the ship’s progress (course and leeway combined). At the upper end of the the same pipe a pointer indicated the amount of leeway. To take into account the effect which change of draught would produce, another pipe was used having the aperture in a neutral direction (41° 30) with regard to the ship’s progress, so that the water was neither· forced in nor drawn out. The two pipes communicated with air-vessels, which were allowed to be about half full of water; thence two flexible tubes conveyed the pressure to the ends of an inverted siphon partly filled with mercury, one leg of which forms a glass index tube, a graduated scale being placed behind it, calculated upon the principle that the pressure will increase according to the square of the velocity. As the specific gravity of mercury is so great, the scale even up to 16 knots is brought within a convenient compass; and it can be hung in gimbals (as a barometer) in any part of the ship. The leeway indicator in more recent fittings has been abandoned. The writer of this article first saw it in one of the Jersey packets, when she was steaming about 13 knots; it appeared to be very sensitive, and he was strongly impressed in its favour. For details respecting this log see paper by Vaughan Pendred, before the Society of Engineers, December 6, 1869.

The motions or disturbances imparted to the water by the body of the ship passing through it at high velocity must vitiate in a great degree all attempts to measure the speed by instruments placed near the hull of the ship, and under varying circumstances of draught, speed, and foulness of bottom. For the results of experiments and opinions on this point, by the late W illiam Froude, F. R. 8., and Mr R. Edmund Froude, see Brit. Assoc. Rep., 1874, p. 225, and 1879, p. 210.

Electric Log.— In the chronological order in which we have taken various descriptions of log, the last deserving notice is Kelway’s “electric log,” the only uch log known to the public. Its chief feature is the making and breaking of an electric circuit by means of a screw revolving in the water aud an electric battery connected with the step motion indicator. One of the difficulties to be overcome was that of securing a chamber wherein to form the electric contacts, which should remain watertight under the pressure due to its depth below the surface of the sea, particularly in the event of the ship stopping and suffering it to sink when being towed with 50 fathoms of line. Mr Kelway now believes that he has overcome that difficulty, and his log has been tried on board several of Her Majesty’s ships at Portsmouth, with satisfactory results, a screw similar to Massey’s being towed, while in electric connexion with a dial on board.

What is considered by Mr Kelway to be an improved application of the principle is now (1882) on view in the International Exhibition at the Crystal Palace. It is intended that a hole should ba cut in the ship’s bottom, by preference in the engine-room, large enough to allow a short cylinder (fig. 5) containing the screw R to pass down below the ship’s bottom. The cylinder is open in a fore-and-aft line and attached to a cage H, which is drawn up or lowered by means of a large screw G working through a stuffing box F. The iron box D containing the cage is 4 feet in height, made in three parts; the lower part (high enough to receive a sluice valve C) is to be bolted to the ship’s bottom, and must, with the rest of the box, be nearly as strong; the central part is secured to the valve box and covered by a lid E, there being space enough above the sluice valve for the cage and screw.

To place the log, let the sluice valve C be lightly closed; open two small taps to let the water out of the box and to prove that the valve is acting. Open the lid, ran the lowering screw through the cage, place it, secure the lid, open the sluice valve, and lower the rotator to the desired distance. The blades of the rotator are portions of a true screw. An endless screw on the spindle of the rotator communicates the revolutions to a vertical spindle M, which moves a train of wheels in a watertight box N; the last of these wheels revolves once in a mile, and on the same spindle is a wheel having eight ratchet teeth, which by moving a lever complete an electric current, which passes by the wire O to a dial placed in any part of the ship, sounding a bell and causing one hand of the dial to make a step and mark an eighth; one revolution indicates a mile, and other dials carry the register up to 100 miles. This form of electric log has, however, the disadvantage pointed out as affecting the Berthon or any log placed under the hull of a ship.

The electric towing log (by Kelway) promises to show continuously on board the ship what she is doing, while keeping a record of what has been done. A rating table would be at the dial, in any part of the ship; or several dials could be worked by the same electric current. It will be exposed to the danger of fouling seaweed, &c., as other towing logs are.

The logs now generally used are Massey’s, Walker ’ sy⅞nd a few of Berthon’s, generally in conjunction with the old log-ship and line. (H. A. Μ.)

Logan was an active member of the committee of the General Assembly of the Church of Scotland appointed in 1775 to revise the “Translations and Paraphrases”· drawn up in 1745 for use in private families, and to adapt them for public worship. The committee finished its; labours in 1781, and their collection of paraphrases is still in use. Eleven of them are the composition of Logan, and others were revised or altered by him. In the same year he published his poems in a volume which attracted so much attention that a second edition was issued in 1782. It also included the “Ode to the Cuckoo,” with which Edmund Burke was so pleased that when in Edinburgh he sought out Logan and complimented him as the author of the finest ode in the English language.

In 1783 he published a tragedy called liunnamede, which met with little success. In 1786 he resigned his charge at South Leith, retaining part of his stipend. He then went to London, where he devoted himself to literature. He was engaged on the management of the English Review, and in 1788 published a pamphlet on the charges against Warren Hastings. He died in December 1788.

A work on ancient history, published, that year under the name of Dr Rutherford, rector of an academy at Uxbridge, is believed to have been the lectures written by Logan. His sermons were published in 1790-91, in two volumes, and have been several times reprinted. His poems were collected and published in 1812, with a memoir understood to be by the Rev. R. Douglas of Galashiels.

About forty years after Logan’s death what may be called the Bruce-Logan controversy arose by the publication in 1836 of a life of Michael Bruce, prefixed to an edition of his poems by the Rev. Dr MacKelvie. In this work there is claimed for Bruce the authorship of sixteen of the pieces in the volume issued by Logan in 1770. Logan was at the same time charged with having retained some of Bruce’s MSS. entrusted to him, which he used in the revision of the paraphrases. These statements have been reiterated with much abuse of Logan in a memoir of Bruce prefixed to an edition of his works published in 1865, by the Rev. Dr Grosart. In this edition the paraphrases written by Logan are inserted as having been written by Bruce. The evidence, however, brought against Logan in these biographies of Brace, being nearly altogether of a hearsay character, is not of much value, and it may be urged that Logan was not blamed during his life for any such literary delinquencies. If anything can be brought against him with justice, it is his publishing as his own, with very few alterations, the second Paraphrase, which is the composition of Dr Doddridge.

Within the last few years the various pieces in the volume of Brace’s poems issued by Logan in 1770 have been subjected to careful criticism, and the statements made from personal knowledge by the Rev. Dr Robertson of Dalmeny, the college friend of Bruce and Logan, who was often referred to on the subject, must be held to be substantially correct. These will be found in a brochure by Dr David Laing, Ode to the Cuckoo (Edinburgh 1770), with remarks on its authorship, in a letter to J. C. Shairp, LL.D., 1873. See also a paper by J. Small in the Brit, and For. Evan. Rev., 1877, and especially two papers by the Rev. R. Small, ibid., 1878.

Logarithms were originally invented for the sake of abbreviating arithmetical calculations, as by their means the operations of multiplication and division may be replaced by those of addition and subtraction, and the operations of raising to powers and extraction of roots by those of multiplication and division. For the purpose of thus simplifying the operations of arithmetic, the base is taken equal to 10, and use is made of tables of logarithms in which the values of x, the logarithm, corresponding to values of m, the number, are tabulated. The logarithm is also a function of frequent occurrence in analysis, being regarded as a known and recognized function like sin x or tan x ; but in mathematical investigations the base generally employed is not 10, but a certain quantity usually denoted by the letter e, of value 2∙71828 18284 ....

Thus in arithmetical calculations if the base is not expressed it is understood to be 10, so that log m denotes log 10 m ; but in analytical formulae it is understood to be e.

The logarithms to base 10 of the first twelve numbers to 7 places of decimals are log 1 = 0 Ό000000 log 5 = 0’6989700 log 9 = 0’9542425

log 2 = 0’3010300 log 6 = 07781513 log 10 = 1'0000000

log 3 = 0’4771213 log 7 = 0∙8450980 log 11 = 1 Ό413927

log 4 = 0’6020600 log 8 ≈ 0'9030900 log 12 = 1’0791812

The meaning of these results is that l = 10° 2 = 10°∙ 3oloaβ ° 3≈10° ' 4771213 · · ·

ιo=ιo 1 , ’ 11-IO 1 · 0413927 ’, 12=IO 1,0791818 ’

The integral part of a logarithm is called the index or characteristic, and the fractional part the mantissa. When the base is 10, the logarithms of all numbers in which the digits are the same, no matter where the decimal point may be, have the same mantissa; thus, for example, log 2∙5613 = 0’4084604, log 25’613 = Γ4084604, log 2561300 = 6‘4084604, &c.

In the case of fractional numbers (i.e., numbers in which the integral part is 0) the mantissa is still kept positive, so that, for example, log ∙25613=T∙4084604, log ∙0025613 = 3∙40846β4, &c., the minus sign being usually written over the characteristic, and not before it, to indicate that the characteristic only and not the whole expression is negative; thus l∙4084604 stands for -1 + ∙4084604.

The fact that when the base is 10 the mantissa of the logarithm is independent of the position of the decimal point in the number affords the chief reason for the choice of 10 as base. The explanation of this property of the base 10 is evident, for a change in the position of the decimal points amounts to multiplication or division by some power of 10, and this corresponds to the addition or subtraction of some integer in the case of the logarithm, the mantissa therefore remaining intact. It should be mentioned that in most tables of trigonometrical functions, the number 10 is added to all the logarithms in the table in order to avoid the use of negative characteristics, so that the characteristic 9 denotes in reality 1, 8 denotes 2, 10 denotes 0, &c. Logarithms thus increased are frequently referred to for the sake of distinction as tabular logarithms, so that the tabular logarithm = the true logarithm + 10.

In tables of logarithms of numbers to base 10 the mantissa only is in general tabulated, as the characteristic of the logarithm of a number can always be written down at sight, the rule being that, if the number is greater than unity, the characteristic is less by unity than the number of digits in the integral portion of it, and that if the number is less than unity the characteristic is negative, and is greater by unity than the number of ciphers between the decimal point and the first significant figure.

It follows very simply from the definition of a logarithm that log α b × log i a = 1, log⅛ m = log β m × —1— .

10g<ι o

The second of these relations is an important one, as it shows that from a table of logarithms to base a, the corresponding table of logarithms to base b may be deduced by multiplying all the logarithms in the former by the constant multiplier —-—, which is called the modulus of log a b the system whose base is b with respect to the system whose base is a.

The two systems of logarithms for which extensive tables have been calculated are the Napierian, or hyperbolic, or natural system, of which the base is e, and the Briggian, or decimal, or common system, of which the base is 10; and we see that the logarithms in the latter system may be deduced from those in the former by multiplication by the constant multiplier 1— which is called the modulus of 1 loge 10

the common system of logarithms. The numerical value of this modulus is 0 - 43429 44819 03251 82765 11289 ..., and the value of its reciprocal, log i 10 (by multiplication by which Briggian logarithms may be converted into Napierian logarithms) is 2 , 30258 50929 94045 68401 79914. . .

The quantity denoted by e is the series, 1 ι i , 1 ι 1 1

1+ 1 + 1.2 + 1.2.3 + 1.2.3.4 + * ' '

the numerical value of which is,

2 ’ 71828 18284 59045 23536 02874 . . .

The mathematical function log x, or log e x, is one of the small group of transcendental functions, consisting only of the circular functions (direct and inverse) sin x, cos x, &c., arc sin x, arc cos x, &c., e x , and log x, which are universally treated in analysis as known functions. It is the inverse of the exponential function e, x , the theory of which may be regarded as including that of the circular functions, since sin x = -{d x -e~ ix ), cos x= ⅛(e i * +e~ ix ). 2λ

There is no series for log x proceeding either by ascending or descending powers of x, but there is an expansion for log (1+ae), viz. :—

log (l+x)=x-^ 2 +⅛≈ 3 -^z 4 + &c.; the series, however, is convergent for real values of x only when x lies between +1 and - 1. Other formulae which are deducible from this equation are given in the portion of this article relating to the calculation of logarithms.

We have also the fundamental formulae—

(i.) Limit of X > when h is indefinitely diminished, = log x;

(h∙) log x+ const.

Either of these results might be regarded as the definition of log x; they may be readily connected with one another, for we have in general

∕∙ Q**∙+l JXl l dx = —— + const.; ∙ ’ n + 1

but if n= -1, this formula no longer gives a result. Putting, however, n = -1 + h, where h is indefinitely small, we have ∕ dx — 1

— = const. =—+ const. = log x + const, by (i.).

The result (ii.) establishes a relation, which is of historical interest, between the logarithmic function and the quadrature of the hyperbola, for, by considering the equation of the hyperbola in the form xy = const., we see at once that the area included between the arc of a hyperbola, its nearest asymptote, and two ordinates drawn parallel to the other asymptote from points on the first asymptote distant α and b from their point of intersection is proportional to log A .

(X>

The function log x is not a uniform function, that is to say, if x denotes a complex variable of the form a + ib, and if complex quantities are represented in the usual manner by points in a plane, then it does not follow that if x describes a closed curve log x also describes a closed curve; in fact we have log (a + iδ) = log √(α 2 + δ 2 ) + i( a +2nιr), where α is a determinate angle, and n denotes any integer. Thus, even when the argument is real, log x has an infinite number of values; for, putting 5 = 0 and taking a positive, in which case α=0, we obtain for log a the infinite system of values log a + 2nπi. It follows from this property of the function that we cannot have’for log x a series which snail be convergent for all values of x, as is the case with sin x, cos x, and e x ; as such a series could only represent a uniform function, and in fact the equation log (l+x)≈x-⅛x 2 +⅛x 3 -⅛x i + &c., is true only when the analytical modulus of x is less than unity.

The notation log x is generally employed in English works, but Continental writers usually denote the function by lx or lgar.

History,— The invention of logarithms has been accorded to John Napier, baron of Mcrchiston, in Scotland, with a unanimity which is rare with regard to important scientific discoveries. The first announcement was made in Napier’s Mirifici logarithmorum canonis descriptio (Edinburgh, 1614), which contains an account of the nature of logarithms, and a table giving natural sines and their logarithms for every minute of the quadrant to seven or eight figures. These logarithms are not what would now be called Napierian or hyperbolic logarithms (f.e., logarithms to the base e), though closely connected with them, the relation between the two being

_ L

e ’ =10 7 e 10’ j or L = 10 7 log e 10 7 -10 7 Z, where l denotes the logarithm to base e and L denotes Napier’s logarithm. The relation between N (a sine) and L its Napierian logarithm is therefore

_ L N = 10 7 e ÏÔ», and the logarithms decrease as the sines increase. Napier died in 1617, and his posthumous work Mirifici logarithmorum canonis constructio, explaining the mode of construction of the table, appeared in 1619, edited by his son.

Henry Briggs, then professor of geometry at Gresham College, London, and afterwards Savilian professor of geometry at Oxford, admired the Canon mirificus so much that he resolved to visit Napier. In a letter to Ussher he writes, “Naper, lord of Markinston, hath set my head and hands at work with his new and admirable logarithms. I hope to see him this summer, if it please God; for I never saw a book which pleased me better, and made me more wonder.” Briggs accordingly visited Napier in 1615, and stayed with him a whole month. He brought with him some calculations he had made, and suggested to Napier the advantages that would result from the choice of 10 as a base, having explained them previously in his lectures at Gresham College, and written to Napier on the subject. Napier said that he had already thought of the change, and pointed out a slight improvement, viz., that the characteristics of numbers greater than unity should be positive and not negative, as suggested by Briggs. In 1616 Briggs again visited Napier and showed him the work he had accomplished, and, he says, he would gladly have paid him a third visit in 1617 had Napier’s life been spared.

Briggs’s Logarithmorum chilias ρrimα was published, probably privately, in 1617, after Napier’s death, as in the short preface he states that why his logarithms are different from those introduced by Napier “sperandum, ejus librum posthumum abunde nobis propediem satisfacturum.” The liber posthumus was the Canonis constructio already mentioned. This wprk of Briggs’s, which contains the first published table of decimal or common logarithms, is only a small octavo tract of sixteen pages, and gives the logarithms of numbers from unity to 1000 to 14 places of decimals. There is no author’s name, place, or date. The date of publication is, however, fixed as 1617 by a letter from Sir Henry Bourchier to Ussher, dated December 6, 1617, containing the passage—“Our kind friend, Mr Briggs, hath lately published a supplement to the most excellent tables of logarithms, which I presume he has sent to you.” Briggs’s tract of 1617 is extremely rare, and has generally been ignored or incorrectly described. Hutton erroneously states that it contains the logarithms to 8 places, and his account has been followed by most writers. There is a copy in the British Museum.

Briggs continued to labour assiduously at the calculation of logarithms, and in 1624 published his Arithmetica logarithmica, a folio work containing the logarithms of the numbers from 1 to 20,000, and from 90,000 to 100,000 (and in some copies to 101,000) to 14 places of decimals. The table occupies 300 pages, and there is an introduction of 88 pages relating to the mode of calculation of the tables, and the applications of logarithms.

There was thus left a gap between 20,000 and 90,000, which was filled up by Adrian Vlacq, who published at Gouda, in Holland, in 1628, a table containing the logarithms of the numbers from unity to 100,000 to 10 places of decimals. Having calculated 70,000 logarithms and copied only 30,000, Vlacq would have been quite entitled to have called his a new work. He designates i⅛ however, only a second edition of Briggs’s Arithmetica logarithmica, the title running Arithmetica logarithmica sive logarith-mοrιι∕m chiliades centum, . . . Editio secunda aucta per Adrianum Vlacq, Goudaηum. This table of Vlacq’s was published, with an English explanation prefixed, at London in 1631 under the title Logarithmicall Arithmetike . . . London, printed by George Miller, 1631. There are also copies with a French title page and introduction (Gouda, 1628).

Briggs had himself been engaged in filling up the gap, and in a letter to Pell, written after the publication of Vlacq’s work, and dated October 25, 1628, he says :—

“My desire was to have those chiliades that are wantinge betwixt 20 and 90 calculated and printed, and I had done them all almost by my seife, and by some frendes whom my rules had sufficiently informed, and by agreement the busines was conveniently parted amongst us; but I am eased of that charge and care by one Adrian Vlacque, an Hollander, who hathe done all the whole hundred chiliades and printed them in Latin, Dutche, and Frenche, 1000 bookes in these 3 languages, and hathe sould them almost all. But he hathe cutt off 4 of my figures throughout; and hathe left out my dedication, and to the reader, and two chapters the 12 and 13, in the rest he hath not varied from me at all.”

The original calculation of the logarithms of numbers from unity to 101,000 was thus performed by Briggs and Vlacq between 1615 and 1628. Vlacq’s table is that from which all the hundreds of tables of logarithms that have subsequently appeared have been derived. It contains of course many errors, which have gradually been discovered and corrected in the course of the two hundred and fifty years that have elapsed, but no fresh calculation has been published. The only exception is Mr Sang’s table (1871), part of which was the result of an original calculation.

The first calculation or publication of Briggian or common logarithms of trigonometrical functions was made in 1620 by Gunter, who was Briggs’s colleague as professor of astronomy in Gresham College. The title of Gunter’s oo , which is very scarce, is Canon triangulorum, and it contains logarithmic sines and tangents for every minute of the quadrant to 7 places of decimals.

The next publication was due to Vlacq, who appended to his logarithms of numbers in the Arithmetica logarithmica of 1628 a table giving log sines, tangents, and secants for every minute of the quadrant to 10 places; these were obtained by calculating the logarithms of the natural sines, &c., given in the Thesaurus Mathematicus of Pitiscus (1613).

During the last years of his life Briggs devoted himself to the calculation of logarithmic sines, &c., and at the time of his death in 1631 he had all but completed a logarithmic canon to every hundredth of a degree. This work was published by Vlacq at his own expense at Gouda in 1633, under the title Trigonometria Britannica. It contains log sines (to 14 places) and tangents (to 10 places), besides natural sines, tangents, and secants, at intervals of a hundredth of a degree. In the same year Vlacq published at Gouda his Trigonometria artificialis, giving log sines and tangents to every 10 seconds of the quadrant to 10 places. This work also contains the logarithms of the numbers from unity to 20,000 taken from the Arithmetica logarithmica of 1628. Briggs appreciated clearly the advantages of a centesimal division of the quadrant, and by dividing the degree into hundredth parts instead of into minutes, made a step towards a reformation in this respect, and but for the appearance of Vlacq’s work the decimal division of the degree might have become recognized, as is now the case with the corresponding division of the second. The calculation of the logarithms not only of numbers but also of the trigonometrical functions is therefore due to Briggs and Vlacq; and the results contained in their four fundamental works— Arithmetica logarithmica (Briggs), 1624; Arithmetica logarithmica (Vlacq), 1628; Trigonometria Britannica (Briggs), 1633; Trigonometria artificialis (Vlacq), 1633—have never been superseded by any subsequent calculations.

A translation of Napier’s Descriptio was made by Edward Wright, whose name is well known in connexion with the history of navigation, and after his death published by his son at London in 1616 under the title A Description of the admirable Table of Logarithmes (12mo); the edition was revised by Napier himself. Both the Descriptio (1614) and the Constructio (1619) were reprinted at Lyons in 1620 by Bartholomew Vincent, who thus was the first to publish logarithms’on the Continent.

Napier calculated no logarithms of numbers, and, as already stated, the logarithms invented by him were not to base e. The first logarithms to the base e were published by John Speidell in his New Logarithmes (London, 1619), which contains hyperbolic log sines, tangents, and secants for every minute of the quadrant to 5 places of decimals.

In 1624 Benjamin Ursinus published at Cologne a canon of logarithms exactly similar to Napier’s in the Descriptio of 1614, only much enlarged. The interval of the arguments is 10", and the‘results are given to 8 places; in Napier’s canon the interval is Γ, and the number of places is 7. The logarithms are strictly Napierian, and the arrangement is identical with that in the canon of 1614. This is the largest Napierian canon that has ever been published.

Kepler took the greatest interest in the invention of logarithms, and in 1624 he published at Marburg a table of Napierian logarithms of sines, with certain additional columns to facilitate special calculations.

The first publication of Briggian logarithms on the Continent is due to Wingate, who published at Paris in 1625 his Arithmétique logarithmétique, containing sevenfigure logarithms of numbers up to 1000, and log sines and tangents from Gunter’s Canon (1620). In the following year, 1626, Denis Henrion published at Paris a Traicté des Logarithmes, containing Briggs’s logarithms of numbers up to 20,001 to 10 places, and Gunter’s log sines and tangents to 7 places for every minute. In the same year De Decker also published at Gouda a work entitled Nieuwe Telkonst, inhoudeηde de Logarithmi voor de Ghetallen beginnende van 1 tot 10,000, which contained logarithms of numbers up to 10,000 to 10 places, taken from Briggs’s...