The Construction of the Wonderful Canon of Logarithms

Front matter

THE

CONSTRUCTION OF LOGARITHMS

WITH A

Preface

THE

CONSTRUCTION OF THE

WONDERFUL CANON OF

LOGARITHMS;

And their relations to their own natural numbers;

WITH

An Appendix as to the making of another and better kind of Logarithms.

TO WHICH ARE ADDED

Propositions for the solution of Spherical Triangles by an

easier method: with Notes on them and on the above-mentioned Appendix by the learned Henry Briggs.

By the Author and Inventor, John Napier, Baron of Merchiston, &c., in Scotland.

Printed by Andrew Hart,

OF EDINBURGH;

in the Year of our Lord, 1619.

Translated from Latin into English by William Rae Macdonald, 1888.

TO THE READER STUDIOUS OF

THE MATHEMATICS,

GREETING.

SEveral years ago (Reader, Lover of the Mathematics) my Father, of memory always to be revered, made public the use of the Wonderful Canon of Logarithms; but, as he himself mentioned on the seventh and on the last pages of the Logarithms, he was decidedly against committing to types the theory and method of its creation, until he had ascertained the opinion and criticism on the Canon of those who are versed in this kind of learning.

But, since his departure from this life, it has been made plain to me by unmistakable proofs, that the most skilled in the mathematical sciences consider this new invention of very great importance, and that nothing more agreeable to them could happen, than if the construction of this Wonderful Canon, or at least so much as might suffice to explain it, go forth into the light for the public benefit.

Therefore, although it is very manifest to me that the Author had not put the finishing touch to this little treatise, yet I have done what in me lay to satisfy their most honourable request, and to afford some assistance to those especially who are weaker in such studies and are apt to stick on the very threshold.

Nor do I doubt, but that this posthumous work would have seen the light in a much more perfect and finished state, if God had granted a longer enjoyment of life to the Author, my most dearly loved father, in whom, by the opinion of the wisest men, among other illustrious gifts this showed itself pre-eminent, that the most difficult matters were unravelled by a sure and easy method, as well as in the fewest words.

You have then (kind Reader) in this little book most amply unfolded the theory of the construction of logarithms, there called by him artificial numbers, for he had this treatise written out beside him several years before the word Logarithm was invented,) in which their nature, characteristics, and various relations to their natural numbers, are clearly demonstrated.

It seemed desirable also to add to the theory an Appendix as to the construction of another and better kind of logarithms (mentioned by the Author in the preface to his Rabdologiæ) in which the logarithm of unity is 0.

After this follows the last fruit of his labours, pointing to the ultimate perfecting of his Logarithmic Trigonometry, namely certain very remarkable propositions for the resolu- tion of spherical triangles not quadrantal, without dividing them into quadrantal or rectangular triangles. These propositions, which are absolutely general, he had determined to reduce into order and successively to prove, had he not been snatched away from us by a too hasty death.

We have also taken care to have printed some Studies on the above-mentioned Propositions, and on the new kind of Logarithms, by that most excellent Mathematician Henry Briggs, public Professor at London, who for the singular friendship which subsisted between him and my father of illustrious memory, took upon himself, in the most willing spirit, the very heavy labour of computing this new Canon, the method of its creation and the explanation of its use being left to the Inventor, Now however, as he has been called away from this life, the burden of the whole business would appear to rest on the shoulders of the most learned Briggs, on whom, too, would appear by some chance to have fallen the task of adorning this Sparta.

Meanwhile (Reader) enjoy the fruits of these labours such as they are, and receive them in good part according to your culture.

Farewell,

Robert Napier, Son.

The Construction

THE CONSTRUCTION OF THE WONDERFUL CANON OF LOGARITHMS; (HEREIN CALLED BY THE AUTHOR THE ARTIFICIAL TABLE) and their relations to their natural numbers.

A

Logarithmic Table is a small table by the use of which we can obtain a knowledge of all geometrical dimensions and motions in space by a very easy calculation.

IT is deservedly called very small, because it does not exceed in size a table of sines; very easy, because by it all multiplications, divisions, and the more difficult extractions of roots are avoided; for by only a very few most easy additions, subtractions, and divisions by two, it measures quite generally all figures and motions.

1. It is picked out from numbers progressing in continuous proportion.

n22. Of continuous progressions, an arithmetical is one which proceeds by equal intervals; a geometrical, one which advances by unequal and proportionally increasing or decreasing intervals.

n33. In these progressions we require accuracy and ease in working. Accuracy ts obtained by taking large numbers for a basis; but large numbers are most easily made from small by adding cyphers.

n44. In computing tables, these large numbers may again be made still larger by placing a period after the number and adding cyphers.

n55. In numbers distinguished thus by a period in their midst, whatever is written after the period is a fraction, the denominator of which ts unity with as many cyphers after it as there are figures after the period.

Thus 10000000.04 is the same as also 25.803 is the same as ; also 9999998 0005021 is the same as , and so of others.

n66. When the tables are computed, the fractions following the period may then be rejected without any sensible error. For in our large numbers, an error which does not exceed unity is insensible and as if it were none.

n77. Besides this, there ts another rule for accuracy; that is to say, when an unknown or incommensurable quantity is included between numerical limits not differing by many units.

7. In place of the unknown quantities themselves, their limits are to be added, subtracted, multiplied, or divided, according as there may be need.

n88. The two limits of one quantity are added to the two limits of another, when the less of the one is added to the less of the other, and the greater of the one to the greater of the other.

n99. The two limits of one quantity are multiplied into the two limits of another, when the less of the one is multiplied into the less of the other, and the greater of the one into the greater of the other.

n1010. Subtraction of limits ts performed by taking the greater limit of the less quantity from the less of the greater, and the less limit of the less quantity from the greater of the greater.

Thus, in the first figure, if from the limits of ac, which are 166.7 and 166.3, you subtract the limits of bc, which are 43.2 and 43.1, the limits of ab become 123.6 and 123.1, and not 123.5 and 123.2, For although the addition of the latter to 43.2 and 43.1 produced 166.7 and 166.3 (as in 8), yet the converse does not follow; for there may be some quantity between 166.7 and 166.3 from which if you subtract some other which is between 43.2 and 43.1, the remainder may not lie between 123.5 and 123.2, but it is impossible for it not to lie between the limits 123.6 and 123.1.

n1111. Division of limits is performed by dividing the greater limit of the dividend by the less of the divisor, and the less of the dividend by the greater of the divisor.

n1212. The vulgar fractions of the limits may be removed by adding unity to the greater limit.

12. Thus far concerning accuracy; what follows concerns ease in working.

n1313. The construction of every arithmetical progression ts easy; not so, however, of every geometrical progression.

13. Those geometrical progressions alone are carried on easily which arise by subtraction of an easy part of the number from the whole number.

n1414. We call easy parts of a number, any parts the denominators of which ave made up of unity and a number of cyphers such parts being obtained by rejecting as many of the figures at the end of the principal number as there are cyphers in the denominator.

n1515. The half, twentieth, two hundredth, and other parts denoted by the number two and cyphers, are also tolerably easily obtained; by rejecting as many of the figures at the end of the principal number as there are cyphers in the denominator, and dividing the remainder by two.

n1616. Hence it follows that if from radius with seven cyphers added you subtract its 100000000th part, and from the number thence arising tts 10000000th part, and so on, a hundred numbers may very easily be continued geometrically in the proportion subsisting between radius and the sine less than it by unity, namely between 10000000 and 9999999; and this series of proportionals we name the First table.

First table.

10000000.0000000

1.0000000

9999999.0000000

.9999999

9999998.0000001

.9999998

9999997.0000003

.9999997

9999996.0000006

to be continued up to

9999900.0004950

Thus from radius, with seven cyphers added for greater accuracy, namely, 10000000.0000000, subtract 1.0000000, you get 9999999.0000000; from this subtract .9999999 you get 9999998.0000001; and proceed in this way, as shown at the side until you create a hundred proportionals, the last of which, if you have computed rightly, will be 9999900.0004950.

n1717. The Second table proceeds from radius with six cyphers added, through fifty other numbers decreasing proportionally in the proportion which zs easiest, and as near as possible to that subsisting between the first and last numbers of the First table.

Second table.

10000000.0000000

100.000000

9999900.0000000

99.999000

9999800.0010000

99.998000

9999700.0001000

99.997000

9999600.006000

Thus the first and last numbers of the First table are 10000000.00000000 and 9999900.0004950, in which proportion it is difficult to form fifty proportional numbers. A near and at the same time an easy proportion is 100000 to 99999, may be continued with sufficient exactness by adding six cyphers to radius and continually subtracting from each number its own 109900th part in the manner shown at the side; and this table

&c up to

9995001.222927

contains, besides radius which is the first, fifty other proportional numbers, the last of which, if you have not erred, you will find to be 9995001.222927.

[This should be 9995001.224804—see note.]

n1818. The Third table consists of sixty-nine columns, and in each column are placed twenty-one numbers, proceeding in the proportion which ts easiest, and as near as possible to that subsisting between the first and last numbers of the Second table.

18. Whence its first column is very easily obtained from radius with five cyphers added, by subtracting tts 2000th part, and so from the other numbers as they arise.

First column of Third table.

10000000.00000

5000.00000

9995000.00000

4995.00125

9985007.49875

4992.50374

9980014.99501

&c up to

9900473.57808

In forming this progression, as the proportion between 10000000.000000, the first of the Second table, and 9995001.222927, the last of the same, is troublesome; therefore compute the twenty-one numbers in the easy proportion of 10000 to 9995, which is sufficiently near to it; the last of these, if you have not erred, will be 9900473.57808.

From these numbers, when computed, the last figure of each may be rejected without sensible error, so that others may hereafter be more easily computed from them.

n1919. The first numbers of all the columns must proceed from radius with four cyphers added, in the proportion easiest and nearest to that subsisting between the first and the last numbers of the first column.

As the first and the last numbers of the first column are 10000000.0000 and 9900473.5780, the easiest proportion very near to this is 100 to 99. Accordingly sixty-eight numbers are to be continued from radius in the ratio of 100 to 99 by subtracting from each one of them its hundredth part.

n2020. In the same proportion a progression ts to be made from the second number of the first column through the second numbers in all the columns, and from the third through the third, and from the fourth through the fourth, and from the others respectively through the others.

Thus from any number in one column, by subtracting its hundredth part, the number of the same rank in the following column is made, and the numbers should be placed in order as follows:—

Proportionals of the Third Table.

First column.

10000000.0000

9995000.0000

9990002.5000

9985007.4987

9980014.9950

&c continuously to

9900473.57808

Second Column.

9900000.0000

9895050.0000

9890102.4750

9885157.4237

9880214.8451

&c continuously to

9801468.8423

Third Column. ‍

9801000.0000

9796099.5000

9791201.4503

9786305.8495

9781412.6967

&c descending to

9703454.1539

Thence 4th, 5th &c., up to

&c., up to

&c., up to

&c., up to

&c., up to

&c., up to

finally to

69th Column. ‍

5048858.8900

5046334.4605

5043811.2932

5041289.3879

5038768.7435

&c descending to

4998609.4034

n2121. Thus, in the Third table, between radius and half radius, you have sixty-eight numbers interpolated, in the proportion of 100 to 99, and between each two of these you have one numbers interpolated in the proportion of 10000 to 9995; and again, in the Second table, between the be two of these, namely between 10000000 an 9995000, you have fifty numbers interpolated in the proportion of 100000 to 99999; and finally, in the First table, between the latter, you have a hundred numbers interpolated in the proportion of radius or 10000000 to 9999999; and since the difference of these is never more than unity, there is no need to divide it more minutely by interpolating means, whence these three tables, after they have been completed will suffice for computing a Logarithmic table.

21. Hitherto we have explained how we may most easily place in tables sines or natural numbers progressing im geometrical proportion.

n2222. It remains, in the Third table at least, to place beside the sines or natural numbers decreasing geometrically their logarithms or artificial numbers increasing arithmetically.

n2323. To increase arithmetiually is, in equal times, to be augmented by a quantity always the same.

Thus from the fixed point b let a line be produced indefinitely in the direction of d. Along this let the point a travel from b towards d, mov- ing according to this law, that in equal moments of time it is borne over the equal spaces b 1, 1 2, 2 3,3 4,4 5 c. Then we call this increase by b 1, b 2, b 3, b 4, b 5, &c, arithmetical. Again, let b 1 be represented in numbers by 10, b 2 by 20, b 3 by 30, b 4 by 40, b 5 by 50; then 10, 20, 30, 40, 50, &c., increase arithmetically, because we see they are always increased by an equal number in equal times.

n2424. To decrease geometrically ts this, that in equal times, First the whole quantity then each of its successive remainders ts diminished, always by a like proportional part.

Thus let the line T S be radius. Along this let the point G travel in the direction of S, so that in equal times it is borne from T to 1, which for example may be the tenth part of T S; and from 1 to 2, the tenth part of 1 S; and from 2 to 3, the tenth part of 2 S; and from 3 to 4, the tenth part of 3 S, and so on. Then the sines T S, 1 S, 2 S, 3 S, 4 S, &c., are said to decrease geometrically, because in equal times they are diminished by unequal spaces similarly proportioned. Let the sine T S be represented in numbers by 10000000, 1 S by 9000000, 2 S by 8100000, 3 S by 7290000, 4 S by 6561000; then these numbers are said to decrease geometrically, being diminished in equal times by a like proportion.

n2525. Whence a geometrically moving point approaching a fixed one has its velocities proportionate to its distances from the fixed one.

Thus, referring to the preceding figure, I say that when the geometrically moving point G is at T, its velocity is as the distance T S, and when G is at 1 its velocity is as 1 S, and when at 2 its velocity is as 2 S, and so of the others. Hence, whatever be the proportion of the distances T S, 1 S, 2 S, 3 S, 4 S, &c., to each other, that of the velocities of G at the points T, 1, 2, 3, 4, &c., to one another, will be the same.

For we observe that a moving point is declared more or less swift, according as it is seen to be borne over a greater or less space in equal times. Hence the ratio of the spaces traversed is necessarily the same as that of the velocities. But the ratio of the spaces traversed in equal times, T 1, T 2, T 3, T 4, T 5, &c., is that of the distances T S, 1 S, 2 S, 3 S, 4 S, &c Hence it follows that the ratio to one ano, of the distances of G from S, namely T S, 1 S, 2 S, 3 S, 4 S, &c., is the same as that of the velocities of G at the points T, 1, 2, 3, 4, &c., respectively.

n2626. The logarithm of a given sine ts that number which has increased avithmetically with the same velocity throughout as that with which radius began to decrease geometrically, and in the same time as radius has decreased to the given sine.

Let the line T S be radius, and d S a given sine in the same line; let g move geometrically from T to d in certain determinate moments of time. Again, let bi be another line, infinite towards i, along which, from b, let a move arithmetically with the same velocity as g had at first when at T; and from the fixed point b in the direction of i let a advance in just the same moments of time up to the point c. The number measuring the line b c is called the logarithm of the given sine d S.

n2727. Whence nothing is the logarithm of radius.

For, referring to the figure, when g is at T making its distance from S radius, the arithmetical point d beginning at b has never proceeded thence. Whence by the definition of distance nothing will be the logarithm of radius.

n2828. Whence also it follows that the logarithm of any given sine ts greater than the difference between radius and the given sine, and less than the difference between radius and the quantity which exceeds it in the ratio of radius to the given sine. And these differences are therefore called the limits of the logarithm.

Thus, the preceding figure being repeated, and S T being produced beyond T to 0, so that 0 S is to T S as T S to d S. I say that b c, the logarithm of the sine d S, is greater than T d and less than 0 T. For in the same time that g is borne from 0 to T, g is borne from T to d, because (by 24) 0 T is such a part of 0 S as T d is of T S, and in the same time (by the definition of a logarithm) is a borne from b to c; so that o T, T d, and b c are distances traversed in equal times. But since g when moving between T and 0 is swifter than at T, and between T and d slower, but at T is equally swift with a (by 26); it follows that 0 T the distance traversed by g moving swiftly is greater, and T d the distance traversed by g moving slowly is less, than b c the distance traversed by the point a with its medium motion, in just the same moments of time; the latter is, consequently, a certain mean between the two former.

Therefore 0 T is called the greater limit, and T d the less limit of the logarithm which b c represents.

n2929. Therefore to find the limits of the logarithm of a given sine.

By the preceding it is proved that the given sine being subtracted from radius the less limit remains, and that radius being multiplied into the less limit and the product divided by the given sine, the greater limit is produced, as in the following example.

n3030. Whence the first proportional of the First table, which is 9999999, has its logarithm between the limits 1.0000001 and 1.0000000.

For (by 29) subtract 9999999 from radius with cyphers added, there will remain unity with its own cyphers for the less limit; this unity with cyphers being multiplied into radius, divide by 9999999 and there will result 1.0000001 for the greater limit, or if you require greater accuracy 1.00000010000001.

n3131. The limits themselves differing insensibly, they or anything between them may be taken as the true logarithm.

Thus in the above example, the logarithm of the sine 9999999 was found to be either 1.0000000 or 1.00000010, or best of all 1.00000005. For since the limits themselves, 1.0000000 and 1.0000001, differ from each other by an insensible fraction like therefore they and whatever is between them will differ still less from the true logarithm lying between these limits, and by a much more insensible error.

n3232. There being any number of sines decreasing from radius in geometrical proportion, of one of which the logarithm or its limits ts given, to find those of the others.

This necessarily follows from the definitions of arithmetical increase, of geometrical decrease, and of a logarithm. For by these definitions, as the sines decrease continually in geometrical proportion, so at the same time their logarithms increase by equal additions in continuous arithmetical progression. Wherefore to any sine in the decreasing geometrical progression there corresponds a logarithm in the increasing arithmetical progression, namely the first to the first, and the second to the second, and so on

So that, if the first logarithm corresponding to the first sine after radius be given, the second ae will be double of it, the third triple, and so of the others; until the logarithms of all sines be known, as the following example will show.

n3333. Hence the logarithms of all the proportional sines of the first table may be included between near limits, and consequently given with sufficient exactness.

Thus since (by 27) the logarithm of radius is 0, and (by 30) the logarithm of 9999999, the first sine after radius in the First table, lies between the limits 1.0000001 and 10000000; necessarily the logarithm of 9999998.0000001, the second sine after radius, will be contained between the double of these limits, namely between 2.0000002 and 2.0000000; and the logarithm of 9999997.0000003, the third will be between the triple of the same, namely between 3.0000003 and 3.0000000. And so with the others, always by equally increasing the limits by the limits of the first, until you have completed the limits of the logarithms of all the proportionals of the First table. You may in this way, if you please, continue the logarithms themselves in an exactly similar progression with little and insensible error; in which case the logarithm of radius will be 0, the logarithm of the first sine after radius (by 31) will be 1.00000005, of the second 2.00000010, of the third 3.00000015, and so of the rest.

n3434. The difference of the logarithms of radius and a given sine is the logarithm of the given sine itself.

This is evident, for (by 27) the logarithm of radius is nothing, and when nothing is subtracted from the logarithm of a given sine, the logarithm of the given sine necessarily remains entire.

n3535. The difference of the logarithms of two sines must be added to the logarithm of the greater that you may have the logarithm of the less, and subtracted from the logarithm of the less that you may have the logarithm of the greater.

Necessarily this is so, since the logarithms increase as the sines decrease, and the less logarithm is the logarithm of the greater sine, and the greater logarithm of the less sine. And therefore it is right to add the difference to the less logarithm, that you may have the greater logarithm though corresponding to the less sine, and on the other hand to subtract the difference from the greater logarithm that you may have the less logarithm though corresponding to the greater sine.

n3636. The logarithms of similarly proportioned sines are equi-different.

This necessarily follows from the definitions of a logarithm and of the two motions, For since by these definitions arithmetical increase always the same corresponds to geometrical decrease similarly proportioned, of necessity we conclude that equi-different logarithms and their limits correspond to similarly proportioned sines. As in the above example from the First table, since there is a like proportion between 9999999.0000000 the first proportional after radius, and 9999997.0000003 the third, to that which is between 9999996.0000006 the fourth and 9999994.0000015 the sixth; therefore 1.00000005 the logarithm of the first differs from 3.00000015 the logarithm of the third, by the same difference that 4.00000020 the logarithm of the fourth, differs from 6.00000030 the logarithm of the sixth proportional. Also there is the same ratio of equality between the differences of the respective limits of the logarithms, namely as the differences of the less among themselves, so also of the greater among themselves, of which logarithms the sines are similarly proportioned.

n3737. Of three sines continued in geometrical proportion, as the square of the mean equals the product of the extremes, so of their logarithms the double of the mean equals the sum of the extremes. Whence any two of these logarithms being given, the third becomes known.

Of the three sines, since the ratio between the first and the second is that between the second and the third, therefore (by 36), of their logarithms, the difference between the first and the second is that between the second and the third. For example, let the first logarithm be represented by the line b c, the second by the line b d, the third by the line b e, all placed in the one line b c d e, thru:—

and let the differences c d and d e be equal. Let b d, the mean of them, be doubled by producing the line from b beyond e to f, so that b f is double b d. Then b f is equal to both the lines b c of the first logarithm and b e of the third, for from the equals b d and d f take away the equals c d and d e, namely c d from b d and d e from d f, and there will remain b c and e f necessarily equal. Thus since the whole b f is equal to both b e and e f, therefore also it will be equal to both b e and b c, which was to be proved. Whence follows the rule, if of three logarithms you double the given mean, and from this subtract a given extreme, the remaining extreme sought for becomes known; and if you add the given extremes and divide the sum by two, the mean becomes known.

n3838. Of four geometrical proportionals, as the product of the means ts equal to the product of the extremes; so of their logarithms, the sum of the means ts equal to the sum of the extremes. Whence any three of these logarithms being given, the fourth becomes known.

Of the four proportionals, since the ratio between the first and second is that between the third and fourth; therefore of their logarithms (by 36), the difference between the first and second is that between the third and fourth. Hence let such quantities be taken in the line b f as that b a

may represent the first logarithm, b c the second, be the third, and b g the fourth, making the dif a c and e g equal, so that d placed in the middle of c e is of necessity also placed in the middle of a g. Then the sum of b c the second and b e the third is equal to the sum of b a the first and b g the fourth. For (by 37) the double of b d, which is b f, is equal to b c and b e together, because their differences from b d, namely c d and d e, are equal; for the same reason the same b f is also equal to b a and b g together, because their differences from b d, namely a d and d g, are also equal. Since, therefore, both the sum of b a and b g and the sum of b c and b e are equal to the double of b d, which is b f, therefore also they are equal to each other, which was to be proved. Whence follows the rule, of these four logarithms if you subtract a known mean from the sum of the known extremes, there is left the mean sought for; and if you subtract a known extreme from the sum of the known means, there is left the extreme sought for.

n3939. The difference of the logarithms of two sines lies between two limits; the greater limit being to radius as the difference of the sines to the less sine, and the less limit being to radius as the difference of the sines to the greater sine.