Page:Encyclopædia Britannica, Ninth Edition, v. 17.djvu/195

OF MERCHISTON,] NAPIER 183 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 calcula tions in 1582. Wittich in 1584 made known at Cassel the calculation of one case by this prosthaphseresis ; and Justus Byrgius proved it in such a manner that from his proof the extension to the solution of all triangles could be deduced. Clavius generalized the method in his treatise De Astroldbio (1593), lib. i. lemma liii. The lemma com mences as follows : " Qu&stiones omnes, qu& per sinus, tancjentcs, atque secantes absolvi solent, per solam prosthaphseresim, id est,per solam additionem, sub- tractionem, sine laboriosa, numerorum multiplicatione divisioneque expedire. "Edidit ante tres quatuorve annos Nicolaus Raymarus Ursus Dithmarsus libellum quendam, in quo prseter alia proponit inven- tum sane acutum, et ingeniosum, quo per solam prosthaphseresjm pleraque triangula sphserica solvit. Sed quoniain id solum putat iieri posse, quando sinus in regula proportionum assumuntur, et sinus totus primum locum obtinet, conabimur nos earn doctrinam magis generalem efficere, ita ut non solum locum habeat in sinibus, et quando sinus totus primum locum in regula proportionum obtinet, verum etiam in tangentibus, secantibus, siuibus versis et aliis numeris, et sive sinus totus sit in principio regulfe proportionum, sive in medio, sive denique nullo modo interveniat : qute res nova oinnino est, ac jucunditatis et voluptatis plena." The work of Kaymarus Ursus, referred to by Clavius, is his Fundamentum Astronomicum (1588). Longomontanus, in his Astronomia Danica (1622), gives an account of the method, stating that it is not to be found in the writings of the Arabs or Regiomontanus. As Longomontanus is mentioned in Anthony Wood s anecdote, and as Wittich as well as Longomontanus were assistants of Tycho, there seems little room for doubt that Wittich s prosthaphseresis is the method referred to by Wood. In 1610 Her wart ab Hohenburg published at Munich a multiplication table extending to 1000 x 1000, a huge folio volume of more than a thousand pages ; and some writers, misled by the title, 1 have supposed that it con tained logarithms. It appears from a correspondence between Kepler and Herwart, 2 which took place at the end of 1608, that Herwart used his table when in manuscript for the performance of multiplications in general, and that the occurrence of the word prosthaphaeresis on the title is due to Kepler, who pointed out that by means of the table spherical triangles could be solved more easily than by Wittich s prosthaphseresis. It is evident that Wittich s prosthaphseresis could not be a good method of practically effecting multiplications unless the quantities to be multiplied were sines, on account of the labour of the interpolations. It satisfies the con dition, however, equally with logarithms, of enabling multiplication to be performed by the aid of a table of single entry ; and, analytically considered, it is not so different in principle from the logarithmic method. In fact, if we put#3/ = <(X + Y), X being a function of x only and Y a function of y only, we can show that we must have X = Ae ?:e , y = Be w ; and if we put #y = <(X + Y) - < (X - Y), the solutions are < (X + Y) = (x + y) 2 , and ?/ = sinY, <(X + Y)= - |cos(X + Y). The 1 Tabulte arithmetic fe Trpocr0a<cupecrec<, s universales, quarum sub- sidio numerus quilibet, ex multipllcoMone produccndus, per solam additionem; et quotiens quilibet, e divisione eliciendus, per solam sub- tractionem, sine tsediosa- < lubrica- Multiplicationis, atque Divisionis operatione, etiam ab eo, qui Arithmetices non admodum sit gnarus, exacte, celeriter & nullo negotio invcnitur. " The correspondence is printed in Frisch s edition of Kepler s works, vol. iv. pp. 527-530. See also a paper "On Multiplication by a Table of Single Entry," in the Philosophical Magazine for November 1878. former solution gives a method known as that of quarter- squares ; the latter gives the method of prosthaphaeresis. An account of the logarithmic table of Justus Byrgius is given in the article LOGARITHMS. The more one considers the condition of science at the time, and the state of the country in which the discovery took place, the more wonderful does the invention of logarithms appear. When algebra had advanced to the point where exponents were introduced, nothing would be more natural than that their utility as a means of performing multiplications and divisions should be remarked ; but it is one of the surprises in the history of science that logarithms were invented as an arithmetical improvement years before their connexion with exponents was known. It is to be noticed also that the invention was not the result of any happy accident. Napier deliberately set himself to abbreviate multiplications and divisions, operations of so fundamental a character that it might well have been thought that they were in reruni natura incapable of abbreviation; and he succeeded in devising, by the help of arithmetic and geometry alone, the one great simplifica tion of which they were susceptible, a simplification to which the following two hundred and seventy years have added nothing. When Napier published the Canon Mirificus England had taken no part in the advance of science, and there is no British author of the time except Napier whose name can be placed in the same rank as those of Copernicus, Tycho Brahe, Kepler, Galileo, or Stevinus. In England, Robert Recorde had indeed published his mathematical treatises, but they were of trifling importance and without influence on the history of science. Scotland had produced nothing, and was perhaps the last country in Europe from which a great mathematical discovery would have been expected. Napier lived, too, not only in a wild country, which was in a lawless and unsettled state during most of his life, but also in a credulous and superstitious age. Like Kepler and all his contemporaries he believed in astrology, and he certainly also had some faith in the power of magic, for there is extant a deed written in his own handwriting containing a contract between himself and Robert Logan of Restalrig, a turbulent baron of des perate character, by which Napier undertakes " to serche and sik out, and be al craft and ingyne that he dow, to tempt, trye, and find out " some buried treasure supposed to be hidden in Logan s fortress at Fastcastle, in considera tion of receiving one-third part of the treasure found by his aid. In the deed Logan also agrees to conduct Napier from Edinburgh to Fastcastle and back again, without his being despoiled of his third part or otherwise harmed, when the deed is to be cancelled and destroyed as a dis charge in full. " And incaiss the said Jhone sal find na poiss to be thair ef tir all tryall and utter diligens tane ; he referris the satisfactione of his trawell and painis to the discretione of the said Robert." Of this singular contract, which is signed "Robert Logane of Restalrige" and "Jhone Neper, Fear of Merchiston," and is dated July 1594, a facsimile is given in Mr Mark Napier s Memoirs? As the deed was not destroyed, but is in existence now, it is to be presumed that the terms of it were not fulfilled ; but the fact that such a contract should have been drawn 3 Of the contract itself Mr Mark Napier writes: "The singularity of his holding conference with one who had just been proclaimed an out law, and whose lawless violence is alluded to and provided against by Napier himself, must be accounted for by the rude state of society, and the simplicity of our philosopher s character. He took care to word the contract itself, however, and there is not an expression which indi cates an idea beyond the most legitimate purpose ; but, under the shield of his own innocence, he never dreamed of contamination from his company, was fond of the romance of science, and not averse (nothing derogatory in his times) to the prospect of gold." Memoirs, p. 223.