Log.

Plate

pinion, is connected with and turns the rod contained in the long copper tube N. This rod, by a pinion fixed at its upper extremity, is connected with and turns upon the whole system of wheels contained in the dial of the case BCD. This dial, by means of the copper tube N, may be fixed to any convenient place aboard the ship. In the front of the dial are several useful circular graduations, as follow: The reference by the dotted line A has a hand which is moved by the wheels within, which points out the motion of the ship in fathoms of 6 feet each. The circle at B has a hand showing the knots, at the rate of 48 feet for each knot: and is to be observed with the halfminute glass at any time. The circle at C has a short and a long hand; the former of which points out the mile in land measure, and the latter or longer the number of knots contained in each mile, viz. 128, which is in the same proportion to a mile as 60 minutes to the hour in the reckoning. At e, a small portion of a circle is seen through the front plate called the register; which shows, in the course of 24 hours (if the ship is upon one tack) the distance in miles that she has run; and in the 24 hours the mariner need take but one observation, as this register serves as an useful ́ check upon the fathoms, knots, and miles, shown upon the two other circles.

North Pole, p. 97. of two other logs, which were tried by Captain Phipps: one invented by Mr Russel, the other by Foxon; both constructed upon this principle, that a spiral, in proceeding its own length in the direc tion of its axis through a resisting medium, makes one revolution round the axis; if therefore the revolutions of the spiral are registered, the number of times it has gone its own length through the water will be known. In both these the motion of the spiral in the water is communicated to the clockwork within board, by means of a small line fastened at one end to the spiral, which tows it after the ship, and at the other to a spindle, which sets the clockwork in motion. That invented by Mr Russel has a half spiral of two threads, made of copper, and a small dial of clockwork, to register the number of turns of the spiral. The other log has a whole spiral of wood with one thread, and a larger piece of clockwork with three dials, two of them to mark the distance, and the other divided into knots and fathoms, to show the rate by the half-minute glass, for the convenience of comparing it with the log. This kind of log will have the advantage of every other in smooth water and moderate weather; and it will be useful in finding the trim of a ship when alone, in surveying a coast in a single ship, or in measuring distances in a boat between headlands and shoals; but it is subject to other inconve niences, which will not render it a proper substitute for the common log.

Perpetual LoG, a machine so called by its inventor, Mr Gottlieb of London, is intended for keeping a constant and regular account of the rate of a ship's velocity in the interval of heaving the log.

Fig. 1. is a representation of the whole machine; CCXCVII. the lower part of which, EFG, is fixed to the side of fig. 1. the keel; H representing only the boundary line of the ship's figure. EF are the section of a wooden external case, left open at the ends KL, to admit the passage of the water during the motion of the ship. At Mis a copper grating, placed to obstruct the entrance of any dirt, &c. into the machine. I is a section of a water wheel, made from 6 to 12 inches in diameter, as may be necessary, with floatboards upon its circumference, like a common water wheel, that turn by the resistance of the water passing through the channel LK. It turns upon a shouldered axis, represented by the vertical section at K. When the ship is in motion, the resistance of the water through the channel LK turns round the wheel I. This wheel, by means of a

f Is a plate showing 100 degrees or 6000 miles, and also acts as another register or check; and is useful in case of any mistake being made in observing the distant run by the other circles. The reckoning by these circles, without fear of mistake, may therefore be continued to nearly 12,000 miles.

A communication from this machine may easily be made to the captain's bedside, where by touching a spring only, a bell in the head ABCD will sound as many times in a half minute as the ship sails miles in an hour.

LOG-Board, a sort of table, divided into several columns, containing the hours of the day and night, the direction of the winds, the course of the ship, and all the material occurrences that happen during the 24 hours, or from noon to noon; together with the latitude by observation. From this table the officers of the ship are furnished with materials to compile their: journals.

Loc-Book, a book into which the contents of the logboard is daily copied at noon, together with every circumstance deserving notice that may happen to the ship, either at sea or in a harbour. See NAVIGATION.

INTRODUCTION.

THE HE labour and time required for performing the arithmetical operations of multiplication, division, and the extraction of roots, were at one time considerable obstacles to the improvement of various branches of knowledge, and in particular the science of astronomy. But about the end of the 16th century, and the beginning of the 17th, several mathematicians be

gan to consider by what means they might simplify these operations, or substitute for them others more easily performed. Their efforts produced some ingenious contrivances for abridging calculations, but of these the most complete by far was that of John Napier Baron of Merchiston in Scotland, who invented a system of numbers called logarithms, which were 30 adapted to the numbers to be multiplied, or divided, that these being arranged in the form of a table, each opposite to the number.

tion.

tion.

2. The difference of the logarithms of any two num- Introducbers, or terms of the geometrical series, is equal to the logarithm of that term of the series which is equal to the quotient arising from the division of the one number by the other.

64 Introduc- number called its logarithm, the product of any two numbers in the table was found by the addition of their logarithms; and, on the contrary, the quotient arising from the division of one number by another was found by the subtraction of the logarithm of the divisor from that of the dividend; and similar simplifications took place in the still more laborious operations of involution and evolution. But before we proceed to relate more particularly the circumstances of this invention, it will be proper to give a general view of the nature of logarithms, and of the circumstances which render them of use in calculation.

Let there be formed two series of numbers, the one constituting a geometrical progression, the first term of which is unity or I, and the common ratio any number whatever, and the other an arithmetical progression, the first term of which is o, and the common difference also any number whatever; (but as a particular example we shall suppose the common ratio of the geometrical series to be 2, and the common difference of the arithmetical series 1), and let the two series be written opposite to each other in the form of a table, thus :

Geom. Prog.

I

16

32

64

128

Arith. Prog.

I

2

3

8623

9

256

512

1024

10

so on.

From the manner in which the two series are related to each other, it will readily appear by induction that the logarithms of the terms of the geometrical series have the two following properties:

1. The sum of the logarithms of any two numbers or terms in the geometrical series is equal to the logarithm of that number, or term of the series, which is equal to their product.

For example, let the terms of the geometrical series be 4 and 32; the terms of the arithmetical series corresponding to them (that is, their logarithms) are 2 and 5; now the product of the numbers is 128, and the sum of their logarithms is 7; and it appears by inspection of the two series, that the latter number is the logarithm of the former, agreeing with the proposition we are illustrating. In like manner, if the numbers or terms of the geometrical series be 16 and 64, the logarithms of which are 4 and 6, we find from the table that 10 4+6 is the logarithm of 1024=16×64; other numbers in the table. and so of any

Take for example the terms 128 and 32, the logarithms of which are 7 and 5; the greater of these numbers divided by the less is 4, and the difference of their logarithms is 2; and by inspecting the two series, this last number will be found to be the logarithm of the former. In like manner, if the terms of the geometrical series be 1024 and 16, the logarithms of which are 10 and 4, we find that 1024÷16-64, and that 10-4-6; now it appears from the table that the latter number, viz. 6, is the logarithm of the former 64.

These two properties of logarithms, the second of which indeed is an immediate consequence of the first, enable us to find with great facility the product or the quotient of any two terms of a geometrical series to which there is adapted an arithmetical series, so that each number has its logarithm opposite to it, as in the preceding short table. For it is evident, that to multiply two numbers we have only to add their loga. rithms, and opposite to that logarithm which is the sum we shall find the product required. Thus, to multiply 16 by 128; to the logarithm of 16, which is 4, we add the logarithm of 128, which is 7, and opposite to the sum 11, we find 2048, the product sought. On the other hand, to divide any number in the table by any other number, we must subtract the logarithm of the divisor from that of the dividend, and look for the remainder among the logarithms, and opposite to it we shall find the number sought. Thus, to divide 2048 by 128; from 11, which is the logarithm of 2048 we subtract 7, the logarithm of 128, and opposite to the remainder 4 we find 16, the quotient sought.

Let us now suppose any number of geometrical means to be interposed between each two adjoining terms of the preceding geometrical series, and the same number of arithmetical means between every two adjoining terms of the arithmetical series; then, as the results will still be a geometrical and an arithmetical series, the interpolated terms of the latter will be the logarithms of the corresponding terms of the former, and the two new series will have the very same properties as the original series.

If we suppose the number of interpolated means to be very great, it will follow that among the terms of the resulting geometrical series, some one or other will be found nearly equal to any proposed number whatever. Therefore, although the preceding table exhibits the logarithms of 1, 2, 4, 8, 16, &c. but does not contain the logarithms of the intermediate numbers, 3, 5, 6, 7, 9, 10, &c. yet it is easy to conceive that a table might be formed by interpolation which should contain, among the terms of the geometrical series, all numbers whatever to a certain extent, (or at least others very nearly equal to them) together with their logarithms. If such a table were constructed, or at least if such terms of the geometrical progression were found together with their logarithms, as were either accurately equal to, or coincided nearly, with all num

bers

tion,

Introduc- bers within certain limits (for example between I and 100000), then, as often as we had occasion to multiply or divide any numbers contained in that table we might evidently obtain the products or quotients by the simple operations of addition and subtraction.

The first invention of logarithms has been attributed by some to Longomontanus, and by others to Juste Byrge, two mathematicians who were cotemporary with Lord Napier; but there is no reason to suppose that either of these anticipated him, for Longomontanus never published any thing on the subject, although he lived thirtythree years after Napier had made known his discovery; and as to Byrge, he is indeed known to have printed a table containing an arithmetical and a geometrical progression written opposite to each other, so as to form in effect a system of logarithms of the same kind as those invented by Napier, without however explaining their nature and use, although it appears from the title he intended to do so, but was probably prevented by some cause unknown to us. But this work was not printed till 1620, six years after Napier had published his dis covery.

It is therefore with good reason that Napier is now universally considered as the first, and most probably as the only inventor. The discovery he published in the year 1614 in a book entitled Mirifici Logarithmorum Canonis Descriptio, but he reserved the construction of the numbers till the opinion of the learned concerning his invention should be known. His work contains a table of the natural sines and cosines, and their logarithms for every minute of the quadrant, as also the differences between the logarithmic sines and cosines, which are in effect the logarithmic tangents. There is no table of the logarithms of numbers; but precepts are given, by which they, as well as the logarithmic tangents, may be found from the table of natural and logarithmic sines.

In explaining the nature of logarithms, Napier supposes some determinate line which represents the radius of a circle to be continually diminished, so as to have successively all possible values, and thus to be equal to every sine, one after another, throughout the quadrant. And he supposes this diminution to be effected by a point moving from one extremity towards the other extremity, (or rather some point very near it), with a motion that is not uniform, but becomes slower and slower, and such, that if the whole time between the beginning and the end of the motion be conceived to be divided into a very great number of equal portions, the decrements taken away in each of these shall be to one another as the respective remainders of the line. According to this mode of conceiving the line to decrease, it is easy to shew that at the end of any successive equal intervals of time from the beginning of the motion, the portions of the line which remain will constitute a decreasing geometrical progression.

Again, he supposes another line to be generated by a point which moves along it equably, or which passes over equal intervals of it in equal times. Thus the portions of the line generated at the end of any equal successive intervals of time from the beginning of the motion will form a series of quantities in arithmetical progression. Now if the two motions be supposed to begin together, at the end of any equal intervals of time the remainders of the one line will form a series of VOL. XII. Part I.

tion.

quantities in geometrical progression, and the corre- Introducsponding portions generated of the other line, will constitute a series in arithmetical progression, so that the latter will be the logarithms of the former. And as the terms of the geometrical progression decrease continually from radius, which is the greatest term, to o, while the terms of the corresponding arithmetical progression increase from o upwards, according to Napier's system the logarithm of radius is o, and the logarithms of the sines from radius down to o, are a series of numbers increasing from o to infinite.

The velocities or degrees of quickness with which the motions commence may have to each other any ratio whatever, and by assuming different ratios we shall have different systems of logarithms. Napier supposed the velocities to be equal; but the system of logarithms produced in consequence of this assumption having been found to have some disadvantages, it has been long disused, and a more convenient one substituted instead of it, as we shall presently have occasion to explain.

Napier's work having been written in Latin was translated into English by Mr Edward Wright, an ingenious mathematician of that period, and the inventor of the principles of what is commonly though erroneously called Mercator's sailing. The translation was sent to Napier for his perusal, and returned with his approbation, and the addition of a few lines, intimating that he intended to make some alterations in the systemi of logarithms in a second edition. Mr Wright die soon after he received back his translation; but it was published after his death, in the year 1616, accompanied with a dedication by his son to the East India Company, and a preface by Henry Briggs, who afterwards distinguished himself so much by his improvement of logarithms. Mr Briggs likewise gave in this work the description and draught of a scale which had been invented by Wright, as also various methods of his own for finding the logarithms of numbers, and the contrary, by means of Napier's table, the use of which had been attended with some inconvenience on account of its containing only such numbers as were the natural sines to every minute of the quadrant and their logarithms. There was an additional inconvenience in using the table, arising from the logarithms being partly positive and partly negative; the latter of these was, however, well remedied by John Speidell in his New Logarithms, first published in the year 1619, which contained the sines, cosines, tangents, cotangents, secants, and cosecants, and given in such a form as to be all positive; and the former was still more completely removed by an additional table, which he gave in the sixth impression of his work, in the year 1624, and which contained the logarithms of the whole numbers 1, 2, 3, 4, &c. to 1000, together with their differences and arithmetical complements, &c. This table is now commonly called hyperbolic logarithms, because the numbers serve to express the areas contained between a hyperbola and its asymptote, and limited by ordinates drawn parallel to the other asymptote. This name, however, is certainly improper, as the same spaces may represent the logarithms of any system whatever, (see FLUXIONS, $152. Ex. 5.).

In 1719 Robert Napier, son of the inventor of lo garithms, published a second edition of his father's Logarithmorum Canonis Descriptio. And along with I

this