182 NAPIER OF MEECHISTON. with great earnestness, he devoted the rest of his life to extend the utility of Napier s splendid invention. Napier s original canon is a table of logarithms of sines, and it was clearly Briggs s original intention to calculate logarithms of sines also ; it does not appear from the account he gives who it was who first suggested the tabulation of the logarithms of numbers instead of sines. Kepler received the invention of logarithms with great enthusiasm. His first mention of them occurs in a letter to Schikhart dated March 11, 1618, in which he writes "Extitit Scotus Baro, cujus nomen mihi excidit, qui prteclari quid praestitit, necessitate omni multiplicationum et divisionum in meras additiones et subtractiones com- mutata, nee sinibus utitur : at tainen opus est ipsi tangentium canone : et varietas, crebritas, difficultasque additionum subtractionumque alicubi laborem multiplicand! et dividend! superat." This erroneous estimate was formed when he had seen the Canon Mirificus but had not read it ; and his opinion was very different when he became ac quainted with the nature of logarithms. The dedication of his Ephemeris for 1620 consists of a letter to Napier dated July 28, 1619, and he there congratulates him warmly on his invention and on the benefit he has conferred upon astronomy generally and also upon his own Rudolphine tables. He says that, although Napier s book had been pub lished five years, he first saw it at Prague two years before; he was then unable to read it, but last year he had met with a little work by Benjamin Ursinus 1 containing the substance of the method, and he at once recognized the importance of what had been effected. He then explains how he verified the canon, and so found that there were no essential errors in it, although there were a few inaccuracies near the begin ning of the quadrant, and he proceeds, " Hsec te obiter scire volui, ut quibus tu methodis incesseris, quas non dubito et plurimas et ingeniosissimas tibi in promptu esse, eas publici juris fieri, mihi saltern (puto et cseteris) scires fore gratis- simum ; eoque percepto, tua promissa folio 57, in debitum cecidisse intelligeres." This letter was written two years after Napier s death, of which Kepler was ignorant, and in the same year as that in which the Constructio was published. In 1624 Kepler published a table of Napierian logarithms, with certain modifications and additions. In a letter from Kepler to Petrus Cugerus there occurs the remarkable sentence " Nihil autem supra Neperianam rationem esse puto : etsi quidem Scotus quidam literis ad Tychonem A. cioioxciv. scriptis jam spem fecit Canonis illius Mirifici." It is here distinctly stated that some Scotsman in the year 1594, in a letter to Tycho Brahe, gave him some hope of the logarithms ; and as Kepler joined Tycho after his expulsion from the island of Huen, and had been so closely associated with him in his work, he would be likely to be correct in any assertion of this kind. In connexion with Kepler s statement the following story, told by Anthony Wood in the Athenee Oxonienses, should be noticed: " It must be now known, that one Dr Craig, a Scotchman . . . coming out of Denmark into his own country, called upon Joh. Neper, Baron of Mercheston, near Edinburgh, and told him, among other discourses, of a new invention in Denmark (by Longomontanus, as tis said), to save the tedious multiplication and division in astro nomical calculations. Neper being solicitous to know farther of him concerning this matter, he could give no other account of it than that it was by proportional numbers. Which hint Nepei taking, he desired him at his return to call upon him again. Craig, after some weeks had passed, did so, and Neper then showed him 1 The title of this work is Benjaminis Ursini . . , Cursus Mathe- matici Practici volumen Primum continens Illustr. & Generosi Dn. Dn. Johannis Neperi Baronis Merchistonij &c. Scoti. Trigonometriam loc/f.rithmicam Usibus discentimn accommodaiam , . . Coloniie . . CIO DC XIX. At the end, Napier s table is reprinted, but to two figures less. As this work was published in 1619, and Vincent s reprint of the Descriplio and Constructio not till 1620, it forms the earliest publication of logarithms on the Continent. a rude draught of what he called Canon mirabilis Logarithmorum. Which draught, with some alterations, he printing in 1614, it came forthwith into the hands of our author Briggs, and into those of Will. Oughtred, from whom the relation of this matter came." Longomontanus Avas Tycho s assistant, and this story, though obviously untrue in its facts, is of importance, as it onnects Dr Craig with Napier and Longomontanus. In the early part of this article Thomas Craig was mentioned as one of the colleagues of Sir ArchiV/aia i-sapier, John Napier s father, in the office of justice-depute. He is well known as the author of a celebrated legal work De Feudis, and between his third son John Craig and John Napier a friendship sprang up which may have been due to their common taste for mathematics. There are extant three letters from Dr John Craig to Tycho Brahe, which show that he was on the most friendly terms with him. In the first letter, of which the date is not given, Craig says that Sir William Stewart has safely delivered to him, " about the beginning of last winter," the book which he sent him. Now Mr Mark Napier found in the library of the univer sity of Edinburgh a mathematical work bearing a sentence in Latin of which the translation is " To Doctor John Craig of Edinburgh, in Scotland, a most illustrious man, highly gifted with various and excellent learning, professor of medicine, and exceedingly skilled in the mathematics, Tycho Brahe hath sent this gift, and with his own hand written this at Uraniburg, 2d November 1588." As Sir William Stewart was sent to Denmark to arrange the pre liminaries of King James s marriage, and returned to Edinburgh on November 15, 1588, there can be little doubt that this was the volume referred to by Craig. It appears from Craig s letter, to which we may therefore assign the date 1589, that, five years before, he had made an attempt to reach Uraniburg, but had been baffled by the storms and rocks of Norway, and that ever since then he had been longing to visit Tycho. Now John Craig was physician to the king, and in 1590 James VI. spent some days at Uraniburg before returning to Scotland from his matrimonial expedition. It seems not unlikely therefore that Craig may have accompanied the king in his visit to Uraniburg. In any case it is certain that Craig was a friend and correspondent of Tycho s, and there can be but little doubt that he was the " Scotus quidam." It is therefore clear that as early as 1594 Napier must have communicated to Craig his hope of being able to effect the simplification of the processes of arithmetic. Everything tends to show that the invention of logarithms was the result of many years of labour and thought, under taken with this special object, and it thus appears that Napier had seen some prospect of success nearly twenty years before the publication of the Canon Mirificus. It is very evident that no mere hint with regard to the use of proportional numbers could have been of any service to Napier, but it is possible that the news brought by Craig of the difficulties placed in the progress of astronomy by the labour of the calculations may have stimulated him to persevere in his efforts. The " new invention in Denmark " to Avhich Anthony Wood refers as having given the hint to Napier was probably the method of calculation called prosthaphseresis (often written in Greek letters Tr/joo-^a^aipecrts), which had its origin in the solution of spherical triangles. 2 The method consists in the use of the formula sin a sin = i{cos(a- &) - cos (a + 5) } , by means of which the multiplication of two sines is 2 A careful examination of the history of the method is given by Scheibel in i$ Einleitung zur matliematischen Buchcrkenntniss, Stiick vii. (Breslau, 1775), pp. 13-29; and there is also an account in Kastner s Oeschichte der Mathematik, vol. i. (1796), pp. 566-569; in Moutucla s Histoire des Malhematiques, vol. i. pp. 583-585 and 617- G19 and in Kliigel s Worterluch (1808), article "Prosthaphseresis."
