John Napier

“ 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. ”

“ The Magnus Canon contains the logarithms of sines for every 10′′ in the quadrant. They are arranged in a similar way, and are of the same kind as those in Napier’s Canon of 1614; but are carried one place
further, radius being made 100,000,000. The entire Canon was recomputed by Ursinus, and full details of its construction are given in Book II., sect. 2, of the Trigonometria. The methods employed are the same as those laid down in the Constructio with the modifications in regard to the preliminary tables proposed by Napier in sect. 60. ”

“ In his preface Mark Napier states that he was induced to publish the work “by the spirited interposition of the Bannatyne and Maitland Clubs of Scotland.” The copies for members of these Clubs are printed entirely in black on their own water-marked paper, size 10 1/2 x 8 1/4 inches. They have not a blank leaf at the beginning, but have on the recto of a leaf after the title-page an extract from the minutes authorising the printing, followed by two leaves containing a list of the members of the Club; the Bannatyne Club having one hundred members and the Maitland Club ninety. ”

“ The signatures in the editions described consist of the letters of the alphabet excluding J, U, and W, or 23 letters (in one or two instances J and U are used for I and V) . To each letter belongs a bundle of leaves, 4, 8, 12, &c., as the case may be. The leaves in each bundle are usually numbered thus :—C, C2, C3, etc., but frequently the signatures are printed only on the first one or two leaves in each bundle. The signature is very rarely printed on a title-page. When a leaf is described as B3, for instance, both sides are included, B31 being used to signify the recto and B32 the verso. ”

“ 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. ”

“ 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 ”

“ Johann Wolther, the Pastor of Liechtenhagen, was a zealous Lutheran, and adherent of the Augsburg Confession. In his work he reprints in full the 36 propositions of Napier’s First treatise as given in the Frankfurt edition of 1615, pp. 1–122, omitting the ‘Beschluz,’ p. 123. To each proposition is appended a refutation of the same. These refutations, being much longer than the propositions, form the bulk of the book. ”

“ These works naturally fall into three groups: the first contains the result of his early studies in Revelation by which he became famous among the Reformed Churches of Europe, as one of the most learned Theologians of the day; another contains his works on Logarithms, by which his fame as a Mathematician was established in the scientific world; between these two groups may be placed his other works, which were more or less preparatory to or suggested during the elaboration of his Logarithms. Accordingly, in the Catalogue we have arranged his works in the following order: I. A Plaine Discovery ”

“ 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 = 1⁄2 {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 ”

“ Accuracy ts obtained by taking large numbers for a basis; but large numbers are most easily made from small by adding cyphers.
⁠Thus instead of 100000, which the less experienced make the greatest sine, the more learned put 10000000, whereby the difference of all sines is better expressed. Wherefore also we use the same for radius and for the greatest of our geometrical proportionals. ”

“ In describing the formation of the Logarithmic Table, in section 59 of the Constructio, Napier says that Reinhold’s common table of sines (or any other more exact) will supply the values for filling in the natural sines in columns 2 and 6, and the table of sines in this work (“Canon Sinvvm Vel Semissivm Rectarvm In Circvlo Svbtensarvm.”, fol 114) , was probably the one he made use of. ”

“ 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. ”

“ 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. ”

“ Following H62 on a folding sheet is the second Table.
This edition contains a translation by Wolffgang Mayer of the two Treatises, but without the Text in the second. All the other matter in the English editions is omitted. The additional matter consists of the Preface and the Alphabetical Index to the principal subjects referred to in the book. The first Table is wanting in the copies both of this edition and that of 1627 in all the libraries noted.
Libraries. Kön. Berlin; Stadt. Breslau; Un. Breslau; Stadt. Frankfurt ”

“ The second and third books of the Trigonometria deal with the subjects treated of in Napier’s Descriptio and Constructio, these works being largely made use of by Ursinus, who speaks of Napier as a Mathematician without equal (see p. 131, l. 5) . ”

“ The special difficulty of this method, however, is in finding the ten proportionals to twelve places by extraction of the fifth root from sixty places, but though this method is considerably more difficult, it is correspondingly more exact for finding both the Logarithms of proportionals and the proportionals of Logarithms. ”

“ 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. ”

“ A possible explanation of this may be that the title-page was originally set up with the date m. dc. xix., but when it was found that the whole work could not be issued in that year, the date was altered to m. dc. x x., and a few copies may have been printed before the alteration. The only copy which we have found is in the Bibliothèque Nationale. The volume contains the three parts; the Tabula has M42 blank; the Constructio has on its title-page the usual date of 1620, and has on H41 the Extraict the same as in the Advocates’ Library copy mentioned in the preceding entry.
Library. Nat. Paris ”

“ 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. ”

“ I. p. 1. At the beginning of the same volume is reprinted the Introduction to Hutton’s Mathematical Tables, on p. liii of which will be found a “brief translation of both parts, omitting only the demonstrations of the propositions, and some rather long illustrations of them.” The logarithms in the Table are of the same kind as Napier’s, but they are not affected by the mistake in the computation of the Canon of 1614. ”

“ 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 , 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. ”

“ For by either method may be had the logarithms of all the proportionals in this table; the last of which is 6934250.8007528, corresponding to the sine 4998609.4034.
47.⁠In the Third table, beside the natural numbers, are to be written their logarithms; so that the Third table, which after this we shall always call the Radical table, may be made complete and perfect. ”

“ The author’s name, it will be observed, is spelt on the title-page in the modern form, and the Dedication to King James is signed, “John Napier, Peer of Marchiston.” The substitution of Peer for Fear or Feuer of Merchiston seems to have been intentional. It is not noticed in the errata, but is of course a mistake.
This is the only edition in which “The Text.”, “The Paraphrasticall Exposition.”, and the “Historicall Application.”, are printed successively and not in parallel columns. The “Historicall Application.”, is printed in black letter. “Az. Chr.” ”

“ 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. ”

“ 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. ”