The Construction of the Wonderful Canon of Logarithms/Notes by the Translator

by John Napier

Notes

by

THE TRANSLATOR

Notes.

Spelling of the Author's Name.

The spelling in ordinary use at the present time is Napier. The older spellings are various—for example, Napeir, Nepair, Nepeir, Neper, Nepper, Naper, Napare, Naipper. Several of these spellings are known to have been used by our author.

I adopt the modern spelling, which is that used by his biographers, and also in the 1645 edition of ‘A Plaine Discovery.’

If, however, the claim of present usage be set aside, a strong case might be made out for Napeir, as this was the spelling adopted in ‘A Plaine Discovery,’ the only book published by Napier in English. In this work is a letter signed “John Napeir” dedicating the book to James VI., and as this letter is a solemn address to the King, we may infer that the signature would be in the most approved form. The work was first issued in 1593, and the same spelling was retained in the subsequent editions during the author's lifetime, as well as in the French editions which were revised by him. In the 1645 edition, as mentioned above, the modern spelling was introduced.

The form Nepair is used in Wright’s translation of the Descriptio, published in 1616, but too much stress must not be laid on this, as very slight importance was attached to the spelling of names; thus although Briggs contributed a preface, his name is spelt in three different ways,—Brigs, Brigges, and Briggs.

In the works published in Latin the form Neperus is invariably used.

On some Terms made use of in the Original Work.

Napier’s Canon or Table of Logarithms does not contain the logarithms of equidifferent numbers, but of sines of equidifferent arcs for every minute in the quadrant. A specimen page of the Table is given in the Catalogue under the 1614 edition of the Descriptio.

The sine of the Quadrant or Radius, which he calls Sinus Totus, was assumed to have the value 100000000.

Numerus Artificialis, or simply Artificialis, is used in the body of the Constructio for Logarithm, the number corresponding to the logarithm being called Numerus Naturalis.

Logarithmus, corresponding to which Numerus Vulgaris is used, is however employed in the title-page and headings of the Constructio, and in the Appendix and following papers. It is also used throughout the Descriptio published in 1614; and as the word was not invented till several years after the completion of the Constructio (see the second page of the Preface, line 12), the latter must have been written some years prior to 1614.

For shortness, Napier sometimes uses the expression logarithm of an arc for the logarithm of the sine of an arc.

The Antilogarithm of an arc, meaning log. sine complement of arc, and the Differential of an arc, meaning log. tangent. of arc (see Descriptio, Bk. I., chap. iii.), are terms used in the original, but as they have a different signification in modern mathematics, we do not use them in the translation.

Prosthapharesis was a term in common use at the beginning of the seventeenth century, and is twice employed by Napier in the Spherical Trigonometry of the Constructio as well as in the Descriptio. The following short extract from Mr Glaisher’s article on Napier, in the ‘Encyclopedia Britannica,’ indicates the nature of this method of calculation.

The “new invention in Denmark” to which Anthony Wood refers as having given the hint to Napier was probably the method of calculation called prosthapharesis (often written in Greek letters προσθαφαιρεσις), which had its origin in the solution of spherical triangles. The method consists in the use of the formula sin a sin b = ½ {cos (a − b) − cos (a + b)}, by means of which the multiplication of two sines is reduced to the addition or subtraction of two tabular results taken from a table of sines; and as such products occur in the solution of spherical triangles, the method affords the solution of spherical triangles in certain cases by addition and subtraction only. It seems to be due to Wittich of Breslau, who was assistant for a short time to Tycho Brahe; and it was used by them in their calculations in 1582.

In the spherical trigonometry the notation used in the original is either of the form 34 gr 24^′ 49^″ or 34: 24^′ : 49^″, but in the translation the form of notation used is always 34° 24′ 49″.

References to delay in publishing the Constructio,
and to a new kind of Logarithms to Base to.

The various passages from Napier’s works bearing on these points are given below.

The first two are referred to by Robert Napier in the first page of the Preface, line 5. They appeared in the Descriptio, published in 1614,—the first, entitled Admonitio, on p. 7 (Bk. I. chap. ii.), and the second, with the title Conclusio, on the 57th or last page of the work (Bk. II. chap. vi.)

The third passage, entitled Admonitio, is printed on the back of the last page of the Table of Logarithms published along with the Descriptio, but is omitted in many copies.

The fourth was inserted by Napier at p. 19 (Bk. I. chap. iv.) of Wright’s translation, published in 1616.

The last is the passage referred to in the second page of the Preface, line 18. It is the opening paragraph in the Dedication of ‘Rabdologize’ to Sir Alexander Seton.

I. From DESCRIPTIO, Book 1. Chapter II.
Note.

Up to this point we have explained the genesis and properties of logarithms, and we should here show by what calculations or method of computing they are to be had. But as we are issuing the whole Table containing the loga- rithms with their sines to every minute of the quadrant, we leave the Theory of their Construction for a more fitting time and pass on to their use. So that their use and advantages being first understood, the rest may either please the more if published hereafter or at least displease the less by being buried in silence, For I await the judgement and criticism of the learned on this before unadvisedly publishing the others and exposing them to the detraction of the envious.

II. From DESCRIPTIO, Book II. Chapter VI.
Conclusion.

It has now, therefore, been sufficiently shown that there are Logarithms, what they are, and of what use they are: for by their help without the trouble of multiplication, division, or extraction of roots we have both demonstrated clearly and shown by examples in both kinds of Trigonometry that the arithmetical solution of every Geometrical question may be very readily obtained. Thus you have, as promised, the wonderful Canon of Logarithms with its very full application, and should I understand by your communications that this is likely to please the more learned of you, I may be encouraged also to publish the method of constructing the table. Meanwhile profit by this little work, and render all praise and glory to God the chief among workers and the helper of all good works.

III. From the End of the TABLE OF LOGARITHMS.
Note.

Since the calculation of this table, which ought to have been accomplished by the labour and assistance of many computors, has been completed by the strength and industry of one alone, it will not be surprising if many errors have crept into it, These, therefore, whether arising from weariness on the part of the computor or carelessness on the part of the printer, let the reader kindly pardon, for at one time weak health, at another attention to more important affairs, hindered me from devoting to them the needful care, But if I perceive that this invention is likely to find favour with the learned, I will perhaps in a short time (with God’s help) give the theory and method either of improving the canon as it stands, or of computing it anew in an improved form, so that by the assistance of a greater number of computors it may ultimately appear in a more polished and accurate shape than was possible by the work of a single individual.

Nothing is perfect at birth.

the end.

IV. From WRIGHT'S TRANSLATION OF THE DESCRIPTIO,
Book I. Chapter IV.
An Admonition.

Bvt because the addition and subtraction of these former numbers [logs. of $\scriptstyle {\frac {1}{10}}$ and its powers] may seeme somewhat painfull, I intend (if it shall please God) in a second Edition, to set out such Logarithmes as shal make those numbers aboue written to fall upon decimal numbers, such as 100,000,000, 200,000,000, 300,000,000, &c., which are easie to bee added or abated to or from any other number,

V. From the DEDICATION OF RABDOLOGIÆ.

Most Illustrious Sir, I have always endeavoured according to my strength and the measure of my ability to do away with the difficulty and tediousness of calculations, the irksomeness of which is wont to deter very many from the study of mathematics, With this aim before me, I undertook the publication of the Canon of Logarithms which I had worked at for a long time in former years; this canon rejected the natural numbers and the more difficult operations performed by them, substituting others which bring out the same results by easy additions, subtractions, and divisions by two and by three. We have now also found out a better kind of logarithms, and have determined (if God grant a continuance of life and health) to make known their method of construction and use; but, owing to our bodily weakness, we leave the actual computation of the new canon to others skilled in this kind of work, more particularly to that very learned scholar, my very dear friend, Henry Briggs, public Professor of Geometry in London.

Notation of Decimal Fractions.

In the actual work of computing the Canon of Logarithms, Napier would continually make use of numbers extending to a great many places, and it was then no doubt that the simple device occurred to him of using a point to separate their integral and fractional parts, It would thus appear that in the working out of his great invention of Logarithms, he was led to devise the system of notation for decimal fractions which has never been improved upon, and which enables us to use fractions with the same facility as whole numbers, thereby immensely increasing the power of arithmetic. A full explanation of the notation is given in sections 4, 5, and 47, but the following extract, translated from ‘Rabdologiæ,’ Bk. I. chap. iv., is interesting as being his first published reference to the subject, though the above sections from the Constructio must have been written long before that date, and the point had actually been made use of in the Canon of Logarithms printed at the end of Wright’s translation of the Descriptio in 1616.

From RABDOLOGIÆ:, Book I, Chapter IV.
Note on Decimal Arithmetic.

The preceeding
example:—divi-
sion by 861094 by
432.

861‍118

86‍141

8‍402

429

861094(1993¹¹⁸⁄₄₃₂

432

3888

8‍3888

86‍1296

861094‍64

86109‍136

8610‍316

861‍118,000

86‍141

8‍402

429

861094,000(1993,273

432

3888

8‍3888

86‍1296000‍

8610‍864

8610‍3024

86109‍1296

But if these fractions be unsatisfactory which have different denominators, owing to the difficulty of working with them, and those give more satisfaction whose denominators are always tenths, hundredths, thousandths, &c., which fractions that learned mathematician, Simon Stevin, in his Decimal Arithmetic denotes thus—①, ②, ③ naming them firsts, seconds, thirds: since there is the same facility in working with these fractions as with whole numbers, you will be able after com- pleting the ordinary division, and adding a period or comma, as in the margin, to add to the dividend or to the remainder one cypher to obtain tenths, two for hundredths, three for thousandths, or more afterwards as required ; and with these you will be able to proceed with the working as above. For instance, in the preceding example, here repeated, to which we have added three cyphers, the quotient will become 1993,273, which signifies 1993 units and 273 thousandth parts or $\scriptstyle {\frac {273}{1000}}$ or, according to Stevin, 1993,2^′7^″3^‴ further the last remainder, 64, is neglected in this decimal arithmetic because 30 ae it is of small value, and similarly in like examples.

Simon Stevin, to whom Napier here refers, was born at Bruges in 1548, and died at The Hague in 1620, He published various mathematical works in Dutch. The Tract on Decimal Arithmetic, which introduced the idea of decimal fractions and a notation for them, was published in 1585 in Dutch, under the title of ‘De Thiende,’ and in the same year in French, under the title of ‘La Disme.’

We find Briggs, in his ‘Remarks on the Appendix,’ while sometimes employing the point, also using the notation 25118865 for $\scriptstyle 2{\frac {5118865}{10000000}}$, distinguishing the fractional part by retaining the line separating the numerator and denominator, but omitting the latter. The form 2|5118865 has also been used. If we take any number such as $\scriptstyle 94{\frac {1305}{10000}}$, the following will give an idea of some of the different notations employed at various times :—

	⓪①②③④
94⓪1①3②0③5④;	941605;

941^′3^″0^‴5^⁗; 941305; 94|1305; 94.1305.

Notwithstanding the simplicity and elegance of the last of these, it was long after Napier’s time—in fact, not till the eighteenth century—that it came into general use.

The subject is referred to by Mark Napier in the ‘Memoirs,’ pp. 451- 455, and by Mr Glaisher in the Report of the 1873 Meeting of the British Association, Transactions of the Sections, p. 16.

On the Occurrence of a Mistake in the Computation of the Second Table; with an Enquiry into the Accuracy of Napier’s Method of Computing his Logarithms.

It is evident that a mistake must somewhere have occurred in the computation of the Second table, since the last proportional therein is given (sec. 17) as 9995001.222927, whereas on trial it will be found to be 9995001.224804.

This mistake introduced an error into the logarithms of the Radical table, as the logarithm of the first proportional in that table is deduced from the logarithm of the last proportional in the Second table by finding the limits of their difference, But these limits are obtained from the proportionals themselves, and, as shown above, one of these proportionals was incorrect: the limits therefore are incorrect, and consequently the logarithm of the first proportional in the Radical table.

We see the effect of this in the logarithm of the last proportional in the Radical table, which is given (sec. 47) as 6934250.8, whereas it should be 6934253.4, the given logarithm thus being less than the true logarithm by 2.6, or rather more than a three millionth part.

The logarithms as published in the original Canon are affected by the above mistake, and also, as mentioned in sec. 60, by the imperfection of the table of sines. It seems desirable, therefore, to enquire whether in addition any error might have been introduced by the method of computation employed.

Before entering on this enquiry, we should premise that in comparing Napier’s logarithms with those to the base e⁻¹ (which is the base required by his reasoning, though the conception of a base was not formally known to him), it must be kept in view that in making radius 10,000,000 he multiplied his numbers and logarithms by that amount, thereby making them integral to as many places as he intended to print. In this we follow his example, omitting, however, from the formula the indication of this multiplication.

In sec. 30, Napier shows that the logarithm of 9999999, the first proportional after radius in the First table, lies between the limits 1.000000100000010 etc., and 1.000000000000000 etc. And in sec. 31, he proposes to take 1.00000005, the arithmetical mean between these limits, as a sufficiently close approximation to the true logarithm; for, the difference of this mean from either limit being .00000005, it cannot differ from the true logarithm by more than that amount, which is the twenty millionth part of the logarithm, But there can be little doubt that Napier was able to satisfy himself that the difference would be very much less, and that his published logarithms would be unaffected.

We proceed to show the precise amount of error thus introduced into the logarithm of 9999999. If we employ the formula

$\log _{e}^{-1}\left(1-{\frac {1}{n}}\right)=\left(-1\right)\left\{-{\frac {1}{n}}-{\frac {1}{2n^{2}}}-{\frac {1}{3n^{3}}}-{\frac {1}{4n^{4}}}-etc.\right\}$

substituting 10000000 for n, and multiplying the result by 10000000, as before explained, we have

1.000000050000003333333583 etc.

Again, if we take the arithmetical mean of the limits, carried to a similar number of places, we have

1.000000050000005000000500 etc.

The error introduced is consequently

.000000000000001666666916 etc.

or about a six hundred billionth part in excess of the true logarithm. It will be observed that besides being very much less, this error is in the opposite direction from that caused by the mistake in the Second table.

We have given above the analytical expression for the true logarithm, namely, $\scriptstyle {\frac {1}{n}}+{\frac {1}{2n^{2}}}+{\frac {1}{3n^{3}}}+{\frac {1}{4n^{4}}}+{\frac {1}{5n^{5}}}+etc.$ The corresponding expression for the arithmetical mean is $\scriptstyle {\frac {1}{n}}+{\frac {1}{2n^{2}}}+{\frac {1}{2n^{3}}}+{\frac {1}{2n^{4}}}+{\frac {1}{2n^{5}}}+etc.$. The latter, therefore, exceeds the true logarithm by $\scriptstyle {\frac {1}{6n^{3}}}+{\frac {1}{8n^{4}}}+{\frac {1}{10n^{5}}}+etc.$, which multiplied by n gives $\scriptstyle {\frac {1}{6n^{2}}}+etc.$ or $\scriptstyle {\frac {1}{6\left(10000000\right)^{2}}}+etc.$, for the error in Napier’s logarithm. So that up to the 15th place the logarithm obtained by Napier’s method of computation is identical with that to the base $\scriptstyle e^{-1}$. If, however, he had used the base $\scriptstyle \left(1-{\frac {1}{n}}\right)^{n}$, where n= 10000000, then the logarithm of 9999999, multiplied by 10000000, as in the other two cases, would necessarily have been unity, or 1.000000000 etc., which would have agreed with the true logarithm to the 8th place only, and would not have left his published logarithms unaffected.

The small error found above in Napier’s logarithm of 9999999 is successively multiplied on its way through the tables: thus, in the First table it is multiplied by 100, in the Second by 50, and in the Third by 20 and again by 69, or in all by 6900000; so that, multiplying the error in the first proportional by that amount, we should have for the error in the logarithm of the last proportional of the Radical table about .0000000115. The error, however, although continually increasing, yet retains always the same ratio to the logarithm, except for a very small disturbing element to be afterwards referred to, so that the true logarithm will always be very nearly equal to the logarithm found by Napier’s method of computation less a six hundred billionth part.

Let us take, for example, the logarithm of 5000000 or half radius. ’ When computed according to Napier’s method, we find it comes out

6931471.80559946464604 etc.

The true logarithm to the base $\scriptstyle e^{-1}$ is

6931471.80559945309422 etc.

So that the difference between the two is

6931471‍.00000001155181 etc.

The six hundred billionth part of the logarithm is

6931471‍.00000001155245 etc.

The latter agrees very closely with the difference found above, and would have agreed to the last place given except for the small disturbing element referred to above, which is introduced in passing from the logarithms of one table to those of the next, or in finding the logarithm of any number not given exactly in the tables as in this case of half radius, but this element is seen to have little effect in modifying the proportionate amount of the original error.

From the above example we see that the error in the logarithm found by Napier’s method amounts only to unity in the 15th place, so that his method of computation clearly gives accurate results far in excess of his requirements, But it is easy to show that Napier’s method may be adapted to meet any requirements of accuracy. In sec. 60, Napier, in suggesting the construction of a table of logarithms to a greater number of places, proposes to take 100000000 as radius. The effect of this would be to throw still further back the error involved in taking the arithmetical mean of the limits for the true logarithm. Thus, using the formula given, substituting 100000000 for n, and multiplying the result by that amount as already explained, we should have for the true logarithm of 99999999, the first proportional after radius in the new First table,

1.000000005000000033333 etc.

If we take the arithmetical mean of the limits, we have

1.000000005000900050000 etc.

This brings out a difference of

.00000000000000016666 etc.,

or a sixty thousand billionth part of the logarithm. We see that the logarithms only begin to differ in the 18th place, and that thus to however many places the radius is taken, the logarithms of proportionals deduced from it will be given with absolute accuracy to a very much greater number of places.

To ensure accuracy in the figures given above, the three preparatory tables were recomputed strictly according to the methods described in the Constructio, fourth proportionals being found in all the preceding tables, and both limits of their logarithms being calculated, the work being carried to the 27th place after the decimal point.

As logarithms to base $\scriptstyle e^{-1}$ are now quite superseded, it is not worth while printing these preparatory tables. The following values (pp. 94-95), however, may be of service for comparison, and as a check to any one who may desire to work out for himself the tables and examples in the Constructio. The values given are the first proportional after radius, and the last proportional in each of the three tables, and also in the Third table, the last proportional in col. 1, and the first proportionals in col. 2 and 69. Opposite these are given their logarithms to base $\scriptstyle e^{-1}$, computed, first, according to Napier’s method, and second, by the present method of series which gives the value true to the last place, which is increased by unit when the next figure is 5 or more. The proportionals and logarithms are each multiplied by 10000000, as explained above.

Though the logarithms in the Canon of 1614 were affected by the

	Proportionals.
First Table.
First proportional after radius,	9999999.
The last proportional,	9999900.000494998383003921217471
Second Table.
First proportional after radius,	9999900.
The last proportional,	9995001.224804023027881398897012
Third Table. Column 1.
First proportional after radius,	9995000.
The last proportional,	9900473.578023286050198667424460
Column 2.
The first proportional,	9900000.
Column 69.
The first proportional,	5048858.887870699519058238006143
The last proportional,	4998609.401853189325032233811730
Half Radius,	5000000.
One-tenth of Radius,	1000000.

mistake in the Second table, this was not the case with those in the Magnus Canon computed by Ursinus and published in 1624. The logarithm of 30° or half radius, for instance, is there given as 69314718 (see specimen page of his Table, given in the Catalogue), which is correct to the number of places given. But in a table of the logarithms of ratios (corresponding to the table in sect. 53 of the Constructio), which is given by Ursinus on page 223 of the ‘Trigonometria,’ the value is stated as 69314718.28, which exceeds the true value by .22. This example will explain how some of the logarithms at the end of the Magnus Canon are too great by 1 in the units place. Notwithstanding

Logarithms computed by Napier's Method.	Logarithms computed by Present Method.
1.000000050000005000000500	1,000000050000003
100.000005000000500000050000	100.000005000000333
100.000500003333525000225002	100.000500003333358
5000.025000166676250011250094	5000.02500016667917
5001.250416822987527739839231	5001.250416822979193
100025.008336459750554796784618	100025.008336459583854
100503.358535014579332632226320	100503.358535014411835
6834228.380380991394618991389791	6834228.380380980004813
6934253.388717451145173788174409	6934253.388717439588668
6931471.805599464646041962236367	6931471.805599453094225
23025850.929940495214660989152136	23025850.929940456840180

this, the Magnus Canon may safely be used to correct the figures in the text and in the Canon of 1614, as the latter is to one place less.

I find no reference by Ursinus to the discrepancies between the logarithms of the two Canons. The mistake in the Second table may possibly not have been observed by him, as the preparatory tables for the Canons were different.

The mistake was observed by Mr Edward Sang in 1865, when recomputing in full the preparatory tables of Napier’s Canon to 15 places.

It had been previously pointed out by M. Biot, in his articles on Napier in the ‘Journal de Savants’ for 1835, p. 255. The following translation of the passage is given in the ‘Edinburgh New Philosophical Journal’ for April 1836, p. 285:—

It has been said, and Delambre repeats the remark, that the last figures of his [Napier’s] numbers are inaccurate: this is a truth, but it would have been a truth of more value to have ascertained whether the inaccuracy resulted from the method, or from some error of calculation in its applications. This I have done, and thereby have detected that there is in fact a slight error of this kind, a very slight error, in the last term of the second progression which he forms preparatory to the calculation of his table. Now all the subsequent steps are deduced from that, which infuses those slight errors that have been remarked. I corrected the error; and then, using his method, but abridging the operations by our more rapid processes of development, calculated the logarithm of 5000000, which is the last in Napier’s table, and consequently that upon which all the errors accumulate; I found for its value 6931471.808942, whereas by the modern series, it ought to be 6931471.805599; thus the difference commences with the tenth figure.

It has been shown in the foregoing pages that the difference referred to does not really commence until the fifteenth figure.

Numerical errata in the text—In consequence of what is mentioned above, the figures in the text are in many places more or less inaccurate, but after careful consideration it is thought that the course least open to objection is to give them as in the original.

Different Methods described in the Appendix for Constructing a Table of Logarithms in which Log. 1=0 and Log. 10=1.

I.

The first method of construction, described on pages 48-50, involves the extraction of fifth roots, from which we may infer that Napier was acquainted with a process by which this could be done. The inference is confirmed by an examination of his ‘Ars Logistica,’ at p. 49 of which (Lib. II., ‘Logistica Arithmetica,’ cap. vii.) he indicates a method by which roots of all degrees may be computed. This method of extraction is referred to by Mark Napier in the ‘Memoirs,’ p. 479 seg., and a translation is there given of the greater part of the chapter above referred to. A method based on the same principles is given by Mr Sang in the chapter “On roots and fractional powers” in his ‘Higher Arithmetic,’ and these principles are also made use of by Mr Sang in his tract on the ‘Solution of Algebraic Equations of all Orders,’ published in 1829.

No general method of extracting roots was known at the time, and it does not appear that Napier had communicated his method to Briggs. At any rate, Briggs did not employ the first method described in computing the logarithms for his canon.

II.

The second method, described on page 51, is a method suitable for finding the logarithms of prime numbers when the logarithms of any two other numbers as 1 and 10 are given. This is done by inserting geometrical means between the numbers, and arithmetical means between their logarithms. The example given is to find the logarithm of 5, but as the example terminates abruptly after the second operation, I append the following table from the article on Logarithms in the ‘Edinburgh Encyclopedia’ (1830), which will sufficiently exhibit the method of working out the example, though it is not carried to the same number of places as that in the text.

THE TABLE.

Numbers.					Logarithms.
A			1.000000		a				0.0000000
B			10.000000		b				1.0000000
C	=	√(AB)	=	3.162277	c	=	⁠1/2⁠(a + b)	=	0.5000000
D	=	√(BC)	=	5.623413	d	=	⁠1/2⁠(b + c)	=	0.7500000
E	=	√(CD)	=	4.216964	e	=	⁠1/2⁠(c + d)	=	0.6250000
F	=	√(DE)	=	4.869674	f	=	⁠1/2⁠(d + e)	=	0.6875000
G	=	√(DF)	=	5.232991	g	=	⁠1/2⁠(d + f)	=	0.7187500
H	=	√(FG)	=	5.048065	h	=	⁠1/2⁠(f + g)	=	0.7031250
I	=	√(FH)	=	4.958069	i	=	⁠1/2⁠(f + h)	=	0.6953125
K	=	√(HI)	=	5.002865	k	=	⁠1/2⁠(h + i)	=	0.6992187
L	=	√(IK)	=	4.980416	l	=	⁠1/2⁠(i + k)	=	0.6972656
M	=	√(KL)	=	4.991627	m	=	⁠1/2⁠(k + l)	=	0.6982421
N	=	√(KM)	=	4.997240	n	=	⁠1/2⁠(k + m)	=	0.6987304
O	=	√(KN)	=	5.000052	o	=	⁠1/2⁠(k + n)	=	0.6987304
P	=	√(NO)	=	4.998647	p	=	⁠1/2⁠(n + o)	=	0.6988525
Q	=	√(OP)	=	4.999350	q	=	⁠1/2⁠(o + p)	=	0.6989135
R	=	√(OP)	=	4.999701	r	=	⁠1/2⁠(o + p)	=	0.6989440
S	=	√(OR)	=	4.999876	s	=	⁠1/2⁠(o + r)	=	0.6989592
T	=	√(OS)	=	4.999963	t	=	⁠1/2⁠(o + s)	=	0.6989668
V	=	√(OT)	=	5.000008	v	=	⁠1/2⁠(o + t)	=	0.6989707
W	=	√(TV)	=	4.999984	w	=	⁠1/2⁠(t + v)	=	0.6989687
X	=	√(VW)	=	4.999997	x	=	⁠1/2⁠(v + w)	=	0.6989697
Y	=	√(VX)	=	5.000003	y	=	⁠1/2⁠(v + x)	=	0.6989702
Z	=	√(XY)	=	5.000000	z	=	⁠1/2⁠(x + y)	=	0.6989700

III.

In the description of the third method, on pages 53-54, it is explained that when log. 1 = 0 and log. 10 is assumed equal to unit with a number of cyphers annexed, a close approximation to the logarithm of any given number may be obtained by finding the number of places in the result produced by raising the given number to a power equal to the assumed logarithm of 10, As an example, Napier mentions that, assuming log. 10 = 1000000000, the number of places, less one, in the result produced by raising 2 to the 1000000000th power will be 301029995. So that reducing these in the ratio of 1000000000, we have log. 10 = 1 and log. 2 =.301029995 &c. The process is explained by Briggs, pages 61-63, and the first steps in the approximation are shown in a tabular form. The table, extended to embrace Napier’s approximation, is given below: in this form it will be found in Hutton’s Introduction to his Mathematical Tables, with further remarks on the subject.

The method, it will be seen, is really one for finding the limits of the logarithm. These limits are carried one place further for each cypher added to the assumed logarithm of 10, but their difference always remains unity in the last place. Bringing together the successive approximations obtained in the table, we find—

When 2 is raised The greater limit of And the less to the power its logarithm is limit is

When 2 is raised to the power	‍	The greater limit of its logarithm is	‍	And the less limit is
1		1.		0.
10		.4		.3
100		.31		.30
1000		.302		.301
10000		.3011		.3010
100000		.30103		.30102
1000000		.301030		.301029
10000000		.3010300		.3010299
100000000		.30103000		.30102999
1000000000		.301029996		.301029995

THE TABLE.

Powers of a.	Indices of powers of a.	Number of places in powers of a.
24619	1000000000‍	1301‍	÷10000‍	=log.	256‍
4	2	1		”„	4
16	4	2		”„	16
256	8	3		”„	256
1024	10	4	÷10	=log.	2
10486 etc.	20	7		”„	4
10995 etc.”	40	13		”„	4
12089 etc.”	80	25		”„	256
12676 &c.	100	31	÷100	=log.	2
16069 &c.”	200	61		”„	4
25823 &c.”	400	121		”„	16
66680 &c.”	800	241		”„	256
10715 &c.”	1000	302	÷1000	=log.	2
11481 &c.”	2000	603		”„	4
13182 &c.”	4000	1205		”„	16
17377 &c.”	8000	2409		”„	256
19950 &c.”	10000	3011	÷10000	=log.	2
39803 &c.”	20000	6021		”„	4
15843 &c.”	40000	120402		&c.
25099 &c.”	80000	24083
99900 &c.”	100000	30103
99801 &c.”	200000	60206
99601 &c.”	400000	120412
99204 &c.”	800000	240824
99006 &c.”	100000	301030
98023 &c.”	2000000	602060
96085 &c.”	4000000	1204120
92323 &c.”	8000000	24082400
90498 &c.”	10000000	3010300
81899 &c.”	20000000	6020600
67075 &c.”	40000000	12041200
44990 &c.”	80000000	24082400
36846 &c.”	100000000	30103000
13577 &c.”	200000000	60206000
18433 &c.”	400000000	120411999
33977 &c.”	800000000	240823997
46129 etc.”	1000000000	301029996