£x Libris 
 K. OCDEN 
 
 THE LIBRARY 
 
 OF 
 
 THE UNIVERSITY 
 
 OF CALIFORNIA 
 
 LOS ANGELES
 
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 NAPIER oS MEIVCMIti TOJ« 
 
 r^t^'f- ff/ /^*^ ^^>^t^>'f/A r»i^, ^i.f_ iTrAi 
 
 A- ^tt^A^^pt.-- >^Cf Afi^ e^ fiu*. Great. Man e,> j>A**>Air /^■*-^/'^/'< 
 
 VIf-4*»rM«i^gft .:: Hume's fiflt. Vi.l.vn, j>. as. .♦ r pdit.i-jiA.
 
 A N 
 
 ACCOUNT 
 
 OF THE 
 
 LIFE, JFRITINGS, and INVENTIONS 
 
 O F 
 
 JOHN NAPIEK, 
 
 O F 
 
 ME R C II IS TON; 
 
 BY 
 
 DAVID STEWART, EARL of BUCHAN, 
 
 AND 
 
 WALTER MINTO, L. L. D. 
 
 ILLUSTRATED iriTH C O P P E R P L yl T E S. 
 
 QUAMDO VI.LUM INVSNIEJ PARFM? 
 
 P E Jl T IT: 
 
 PRIXTED Br R. MORISOX, yUXR. 
 
 FOR R. MORISON AND SON, nOOKSEI.I.ERS ; AND SOI.n BY G. (J. .1. 
 
 AND J. ROBINSON, PATER-NOSTE R- ROW, LONDON; 
 
 AND W. CREECH, EDINBURGH. 
 
 M,DCC,LXXXVII.
 
 
 T O T 11 £ 
 
 KING. 
 
 SI R, 
 
 As the writings of Archimedes were addrened to the King- of Sicily, 
 who had perufed and rehllied them, fo I do myfelf the honour, to ad- 
 drcfs to Your Majefty, the following account of the Life, Writings, 
 and Inventions of bur Britilh Archimedes, in which, I can claim no 
 other merit, than having endeavoured to call forth and illuftrate tlic 
 abilities of others. I feel great plcafurc, in dedicating this TraCi to 
 Your Majefty, after the chafte and dignified model of Antiquity, be- 
 ftowing on the King, the merited encomium, of having promoted the 
 Sciences and Arts, with which it is connccled ; and in alTuring Your 
 Majefty, that I am, with the greatcft refpcct, 
 
 Your Majesty's 
 
 Moll dutiful Subjed, and 
 
 Obedient humble Servant, 
 
 li U C H A N. 
 
 lorBO^*^
 
 ''iitif==s=r=>s£^iX7i ^'^'^<r ' . i La ::g= 
 
 
 ADVERTISEMENT. 
 
 tyrLDOUT twenty years ago, I thought it would be cafy to bring together a groupe 
 of learned men, who would dedicate a part of their leifure to ered literary monuments 
 to the memory of their illujlrious countrymen, whofe lives had not been hitherto 
 written or fufficiently illujirated ; and Iwifhed fuch monuments to he fajhioned and ex- 
 ecuted by men perfonally eminent in the departments which dijlinguipcd the fuhjcds of 
 their biographical rcfearch, and not by the ajfijlants of a bookfclkr or compiler, who 
 cannot be expelled, however faithful and accurate, to be animated with that love to the 
 fuhje6l, which the Italian Artiji rightly confiders as the foul of his enterprise, and 
 the fource of its perfcdion. 
 
 In this expeSlation I have been difappoinled ; and though I allow the highejl mcrtt 
 to the Britijh Biography, now republijhing by Dr Kippis, yet in the immenfe extent of 
 fuch an undertaking, I perceive the impojibility of its reaching the perfeBion I have 
 propofed, without the addition of fupplementary articles and connexions, which would 
 have been in a great meafure unneceffary, had my plan been adopted; bccaufe the ar- 
 ticles, being written with care and with zeal, fo as to fupport thcmfclves in an ifo- 
 latedflate by the public favour, would afterwards have been taken up by fubfequtnt 
 editions into that great repofitory of biographical learning, in a highly fnijlied flate, 
 and purged of the errors which are unavoidable, in the firfl fabric of works of that 
 
 nature. 
 
 b With
 
 [ vi ] 
 
 iTith rej^c^ lo the biography of Scotland, one of the judges there, who would have 
 done it honour in its bcji days, by his virtue, his attention to the dignity and duties of 
 hisjlation, and the tfcful employment cf his leifure, has generoufly o^ered, by an ad- 
 vert ifement annexed to the Annals of the Lives of fehn Barclay, Author of Arrenis, 
 and fome other learned Scots, to forward the undertaking I wiJJj to promote. 
 
 Encouraged by the ajfflance of an affociato, fa able andfo liberal, I have prefumed 
 to offer the following Biographical Trad to the public, as my mite to a Treafury, which 
 I hope to fee enriched by many, who have the ability and the generofity of my refpedable 
 coadjutor. It was indeed by that excellent man, that 1 was originally encouraged to 
 profecute refearches of this nature. He applauded that difpofttion in a young man cf 
 quality, which leads him to the fcudy of the hiflory of his own country, not in pamphlets, 
 fatires, apologies and panegyrics, but in the private undifguifed correfpondence of the 
 great. 
 
 A man who fludies hiflory in this way, will fee that the fame charaSlcrs are re~ 
 prefcnted by different aElors : introduced behind the fcenes, he will fee folly drefftng 
 itfelf in the garb of wifdom, and felfijhnefs affuming the mafk of public fpirit ;• and 
 among the learned, the plagiary flealing away the laurels of the modefi inventor. He 
 will fee great events ariftng from inconftderable caufes, and men neither devils nor 
 angels, but a compofttion of good and bad qualities, fuch as the men of the world can 
 fee them every day in common life. 
 
 1 flatter myfelf, that this article of Napier, in the Biographia Scotica, will be conjider- 
 ed in fome refpecls, as afpecimen of the plan I have defcribed, for it certainly has been 
 written con amore. In the fcientifc part I have received the affiflance of a gentle- 
 man, who deferves to be better known, on account of his mathematical learning, and 
 the accuracy with which he treats the fuhjeils of his inquiry. 
 
 If.
 
 [ vli ] 
 
 If the following publication, ^mll have the good fortune to meet with the approbation 
 of the learned world, 'tis my intention, to give an account of the lives and writings of 
 Andrew Fletcher of Salt on, and fohn Law of Lauricjlon, on the fame plan. The 
 firjl undertaking will fur nijli me an opportunity, of reprefcnting the ancient confiituticn 
 of Scotland, in what I apprehend to be a clearer light, than has hitherto been offered ; 
 and of treating the caufes and confequences of the union between tbe two kingdoms : 
 and the other will open an ample field for exhibiting the diforders in the finances of 
 France, occafioned by the expenfive wars of Lewis the fourteenth, and the Mifjifippl 
 Scheme, and for explaining by what means they have been gradually remedied and 
 brought to a flatc, which has enabled that nation, not only to bring her naval force 
 and her trade to a dangerous rivaljlnp with this country, but to obtain that credit, 
 by good faith, which informer times, had given fo decided a fuperiority to Britain. I 
 am very fenftble that there are many men in this country much better qualified for 
 performing thefe tafks than I am, and I think it an honour to enjoy their friendjl/ip : 
 but men of great reputation generally feek for refl in the evening of life, and avoid cs- 
 pofing their laurels to the blajl of envy, in their declining years. 
 
 Thefe, I hope, will be accepted as fifficitnt apologies, for my venturing to occups fuch 
 ground, and I beg leave to invite my learned countrymen, to aid me in fo noble an un- 
 dcrtaking, as that of raifing monuments to the memory of the illuflrious dead. 
 
 / have only to add, that if the feparate lives of illujirious perfons, Jlootdd be written 
 on the plan I prcpofe, and were accompanied by portraits, elegantly engraven by the 
 bcjl artifls, and the whole executed in afimilar manner, of the fame ^lartofize, and 
 with the fame Type and Paper, they would gradually form the nobleft work, which has 
 been offered to the republic of letters, in any age or country. 
 
 AN
 
 A N 
 
 ACCOUNT 
 
 or THE 
 
 LIFE, WRITINGS, and INVENTIONS, 
 
 O F 
 
 JOHN NAPIER. 
 
 O F 
 
 MERCHISTON. 
 
 X Have xmdertaken to write the Life of John Napier, of Mercliifton, 
 a man famous all the world over, for his great and fortunate difcove- 
 ry of Logarithms in Trigonometry, by which the cafe and expedition 
 in calculation, have fo wonderfully aflifled the Science of Aftrononiy, 
 and the arts of pra<5lical Geometry and Navigation. 
 
 Elevated above the age in which he lived, and a benefaclor to the 
 world in general, he defcrves the epithet of Great. 
 
 Napier lived in a country of proud Barons, where barbarous hofpi- 
 tality, hunting, the mihtary art, and religious controvcrfy, occupied 
 
 G the
 
 lo LIFEofNAPIER. 
 
 the time and attention of his contemporaries, and where he had no 
 learned fociety to aflift him in his rel'earches. 
 
 This extraordinary perfon was born at Merchifton, in the neigh- 
 bourhood of Edinburgh, in the year 1550*. 
 
 He was the Son of Sir Archibald Napier, of Mcrchillon, Mafter of 
 the Mint in Scotland, and of Janet Bothwell, daughter of Mr Francis 
 Botliwell, one of the Senators of the college of Juftice f . 
 
 That his family was of ancient eflablifhment in the counties of 
 Dunbarton and Stirling, appears from the public records, and from the 
 private archieves of his houfe. 
 
 John de Napier, from whom he fprung in the 12th generation, was 
 one of thofe proprietors of lands, who fwore allegiance to Edward the 
 firft, of England, in the year 1296. William, from whom he count- 
 ed in the ninth generation, was Governor of the Caftle of Edinburgh, 
 in the year 1401, whofe fon Alexander, was the firft Baron or Laird of 
 Merchifton, and was die Father of another of the fame name, who was 
 Vice Admiral of Scotland, and one of the Conuniffioners from king 
 James the third, at the court of London, in the years 1461 and 1464. 
 
 From the family of Lennox, Earl of Lennox, he derived a coheirfhip 
 by the marriage of Elizabeth Mentieth, of Ruilcy, to his great-grand- 
 father's 
 
 • As appears by an infcription on his portrait, engraved by old Cooper, from an original painting. 
 t Craufurd'b Peerage.
 
 LIFE OF NAPIER. 
 
 II 
 
 father's father, Sir John Napier, of Merchifton : but on his anceftors 
 he reflected more honour and celebrity than he received, and his name 
 will probably be famous, when the lineage of Plantagenet will be re- 
 membred only by genealogifts, and when pofterity may know no more 
 of his, than we now know of the families of Plato, Ariftotle, Archi- 
 jinedes, or Euclid. 
 
 It is fit, that men fliould be taught to aim at higher and more per- 
 manent glory than wealth, office, titles or parade can afford ; and I like 
 the talk, of making fuch great men look little, by comparing them 
 with men who refemble tlie fubjecl of my prefent enquiry. 
 
 From Napier's own authority, we learn, that he was educated at St. 
 Andrews *, where writes he, " in my tender years and bairn-age, at 
 " fchools, having on the one part contracled a loving famiharitie with 
 *' a certain gentleman a papift, and on the other part being attentive to 
 *' the fermons of that worthy man of God, Maifter Chriftopher Good- 
 " man, teaching upon the Apocalyps, I was moved in admiration againft 
 " the bhndnefs of papifts that could not moft evidentlie fee their feven 
 " hilled Citie of Rome, painted out there fo lively by Saint John, as the 
 " Mother of all Spiritual Whoredome : that not onlie burfted I oute in 
 " continuall reafoning againft my faid familiar, but alfo from thence- 
 " forth I determined with myfelf by the affiftance of God's fpirit to 
 " employ my ftudy and diligence to fearch out the remanent myfterics 
 " of that holy booke (as to this houre praifed be the Lord I have bin 
 " doing at all fuch times as convenientlie I might liave occafion) tsfc. 
 
 TuE 
 
 • Preface to lib plain diTcovcry of th« Revelation of St. John.
 
 12 L I F E OF N A P I E R. 
 
 The time of Napier's matriculation does not appear from the Regi- 
 fter of the Univerfity of St. Andrews, as the books afcend no higher 
 than the beginning of the laft century ; but as the old whore of Baby- 
 lon, affumed in the eyes of the people of Scotland, her deepefl tinge of 
 fcarlet about the year 1566, and as that time corrcfponds to the literary 
 bairn-age of John Napier, I fuppofe, he then imbibed the holy fears 
 and commentaries of Maifter Chriftopher Goodman, and as other 
 great Mathematicians have ended, fo he began his career with that 
 ni}'flerious book. 
 
 I have not been able to trace Merchiflon from the Univei-fity, till the 
 publication of liis Plain Difcovery, at Edinburgh, in the year 1593*; 
 though Mackenzie in his lives and chara(5lers of the moft eminent wri- 
 ters of the Scotifh nation, informs us (without quotation, however, of 
 any authority) that he pafTed fome years abroad, in the low countries, 
 France and Italy, and that he applied himfelf there, to the ftudy of 
 Mathematics. 
 
 In the Britifla Mufeum there are two copies of liis letter to Anthony 
 Bacon, the original of which, is in the Archbifhop's Library at Lam- 
 beth, entitled " Secret Inventions, profitable and neceflar)^, in thefe 
 days, for the defence of this Ifland, and withftanding ftrangers enemies 
 to God's truth and religion," which I have caufed to be printed, in the 
 Appendix to this Tradl. Tliis letter is dated, June 7, 1596 j", about 
 which time it appears, as (hall be fhewn hereafter, that he had fet him- 
 felf to explore his Logarithmic Canon. 
 
 I 
 
 • Piintcd by Andrew Hart, 410, t Ayfcongles lat. r. i. p. 15J. See Appendix N". i.
 
 L I F E OF N A P I E R. 13 
 
 I have enquired, witliout fuccefs, among all the defcendants of this 
 eminent perfon, for papers or letters, which might elucidate this dark 
 part of his hiftory ; and if we confider that Napier was a reclufe mathe- 
 matician, living in a country, very inacceflible to literary correfpon- 
 dence, we have not much room to expedt, that the moft diligent explo- 
 rations, would furnifli much to the purpofe, of having the progrefs of 
 his ftudies. 
 
 Among Mr Briggs's papers, preferved in the Britifli Mufeum, I look- 
 ed for letters from Napier, but found only what Mr Briggs calls, his 
 Imitatio Nepeirea,Jive applicat'to omnium fere regiilarum^fiiis Logarithmis per- 
 tinentitim, ad Logaritbmos ; which feems to have been written in the year 
 1614, foon after the publication of the Canon*. 
 
 Though the life of a learned man is commonly barren of events, 
 and beft xmfolded in the account of his writings, difcoveries, improve- 
 ments, and corrcfpondence with the learned men of his ago, yet I 
 anxioufly fought for fomewhat more, with refpedl to a character, I fo 
 much admired ; but my refearches have hitherto been fruitlefs. Per- 
 haps from the letters, boots, and colleflions of focieties or of learned 
 individuals, to which I have not had accefs, fomething may hereafter be 
 brought to light : and one of the inducements, to offer a flcetch of this 
 kind to the public, is the tendency it may have to bring forth fuch in- 
 formation. His plain difcovery has been printed abroad, in fcveral lan- 
 guages, particularly in French, at Rochelle, in the year 1603, 8vo. an- 
 
 D nounced 
 
 • A)-fcoiigh'6 Cat. vol. 1. p. 389.
 
 ,4 L I F E OF N A P I E R. 
 
 nounced in the tide, as revifed by liimfelf *. Nothing could be more 
 agreeable to the Rochellers, or to the Hugonots of France, at tliis time, 
 than the Author's annunciation of the Pope, as Antichrill, wliich in thii 
 book he has endeavoured to fet forth, witli much zeal and erudition. 
 
 That Napier had begun, about tlie year 1593, that train of enquiry, 
 which led him to his great atchievement in Arithmetic, appears from 
 a letter to Crugerus from Kepler, in the year 1624; wherein, menti- 
 oning the Canon Mirificus, he writes thus. Nihil aiUcmfupra Neperianam 
 ratiouem ejfe puto: ctfi Scotus quidam Uteris ad TycboneiTiy anno 1594, Scriptis 
 jamfpcmfecit Canonis illitis mirifici, which allufion agrees with the idle ftory 
 mentioned by Wood in his Athenx Oxon, and explains it in a way 
 perfectly confonant to the rights of Napier as the inventor ; concerning 
 which, I fliall take occafion to comment, in the account of his works : 
 nor is it to be fuppofed, that had this noble difcovery been properly 
 applied to fcience, by Jufhus Byrgius, or Longomontanus, Napier would 
 have been univerfally acknowledged by his contemporaries, as the un- 
 difputed Author of it. 
 
 No men in the world, are fo jealous of each other as the learned, and 
 the leafl plaufible pretence of this fort, could not have failed to produce 
 
 a 
 
 * This edition was publifhed on the firft day of that year, in the end of which the Synod of Gap 
 did declare, or moved to declare, the Pope to be Antichrift, which liad never been before attempted, 
 by any body of Proteftants. See Sully's Memoirs. 
 
 With refpeft to Napier's fanciful calculation of the completion of the prophecies, concerning the 
 duration of the world, the year, ia which this monument is erefted to his memory, immediately fuc- 
 ceeds that fixed for the end of the world, and no doubt muft be the year of judgment, with refpeft to 
 tjie authenticity of his difcovery, and the merit of thofe arguments, which are brought forward to 
 fupport his claim..
 
 L I F E o F N A P I E R. 15 
 
 a controverfy, in the republic of letters, both in his llfetin\:, and after 
 his deatli, when his praifes were {bunded all over Europe *. 
 
 When 
 
 • To quote authorities in this place, would be to give a catalogue of all the Mathematical aod 
 Arithmetical books of that age. 
 
 His moft outrageous pancgyrift, is Sir Thomas Urquhart, of Cromaity, who has given us alfo fo 
 ridiculous an account of the admirable Crichton. 
 
 In his Jewel, Urquhart, after having referred his readers to his Trigonometrical Work, entitled 
 Triffotelras, for the praifes of Napier, thus mentions " an almoft incomprehenfible device, which bc- 
 " in.T in the mouths of the moft of Scotland, and yet unknown to any that ever was in the world but 
 " himfclf, deferveth very well to be taken notice of in this place ; and it is this : he had the /l:il], at 
 " is commonly reported, to frame an engine, (for invention not much unlike that of Archylcas's Dove) 
 " which by virtue of fome fecret fprings, inward refforts, with other im.plements, and materials Gt for 
 " the purpofe, inclofed witliin the bowels thereof, had the power (if proportionable in bulk to the 
 " adlion required of it (for he could have made it of all fizes) to clear a field of four miles circum- 
 " ference, of all the living creatures exceeding a foot in heighth, that (liould be found thereon, how 
 " near foever they might be found to one another ; by which means he made it appear, that he wag 
 " able, with the help of this machine alone, lo kill thirty thoufand Turks, without the hazard of one 
 " Chriftian ! " Of this it is faid that (on a wager) he gave proof upon a large plain in Scotland, to 
 the deftrudlion of a great many head of Cattle, and flocks of (heep, whereof fome were diftant from 
 other half a mile on all fides, and fome a whole mile. To continue the thread of my ftory, as I have it, 
 I mud not forget, that when he was moft eameftly defired by an old acquaintance, and profcffed friend 
 of his, even about the time of his contraAing the difeafe whereof he died, that he would be pleafcd, for 
 the honour of his family, and his own evcrlafting memory to poftcrity, to reveal unto him the manner 
 of the contrivance of fo ingenious a myftcry, fubjoining thereto, for the better perfuading him, that it 
 were a thoufand pities, that fo excellent an Invention (hould be buried with liim in the grave, and that 
 after hisdcceafe nothing (hould be known thereof: his anfwer was, that for the ruin and overthrow 
 of man, there were too many devices already framed, which if he could make to be fewer, he would 
 with all his might endeavour to do ; and that, therefore, feeing the maL'ce and rancor rooted in the 
 heart of mankind, will not fuffer them to diminifti the number of them, by^any new concert of hit 
 (hould never be incrcafed. Divinely fpoken truly. 
 
 Urquhart's Tradls, Edinburgh, 1774. 8vo. p. 57.
 
 :6 L I F E OF N A P 1 E R. 
 
 When Napier had communicated to Mr Henry Brlggs, Mathemati- 
 cal ProfefTor in Grelham College, his wonderful Canon for the Loga- 
 rithms, that learned ProfefTor fct himfelf to apply the rules in his ////;'- 
 tatlo Nepeirea, which I have already mentioned, and in a letter to 
 Archbifhop Ufher, in the year 1615, he writes thus, " Napier, Lord of 
 " Merchifton, hath fet my head and hands at work with his new and 
 " admircable Logarithms. I hope to fee him this fummer if it pleafe 
 " God, for I never faw a book which pleafed me better, and made mc 
 " more wonder'*. 
 
 It may feem extraordinary to quote Lilly the aftrologcr witli refpedl 
 to fo great a man as Napier; yet as the pafTage I propofe to tranfcribe 
 from Lilly's life, gives a pi(5lurefque view of tlae meeting betwixt Briggs 
 and the Inventor of the Logarithms, at Merchiilon near Edinburgh, 
 I fliall fet it down in the original words of that mountebank knave f . 
 
 *' I will acquaint you with one memorable ftory, related unto me 
 by John Marr, an excellent mathematician and geometrician, whom I 
 conceive you remember. He was fcrvant to king James the firfl and 
 Charles the firft. When Merchiflon firft publiflied his Logarithms, 
 Mr Briggs then reader of the Aftronomy Lecflures at GrcHiam College 
 in London, was fo furprifed with admiration of them, that he could 
 have no quietnefs in himfelf, until he had feen that noble perfon whofe 
 only invention they were : He acquaints John Marr therewith, who went 
 into Scotland before Mr Briggs, ptirpofely to be there when thefe two 
 fo learned perfons fliould meet ; Mr Briggs appoints a certain day when 
 
 to 
 
 • Uihcr's Letters, p. 36. t Lilly's Life, London, 1721. 8vo.
 
 LIFEofNAPIER. 17 
 
 to meet at Edinburgh, but failing thereof, Merchifton was fearful he 
 would not come. It happened one day as John Marr and the Lord 
 Napier were fpeaking of Mr Briggs ; " Ah John, faith Merchifton, Mr 
 Briggs will not now come" : at the very inftant one knocks at the gate ; 
 John Marr hafted down and it proved to be Mr Briggs to his great con- 
 tentment. He brings Mr Briggs up into My Lord's chamber, where 
 almoft one quarter of an hour was fpent, each beholding other with ad- 
 miration before one word was fpoken : at laft Mr Briggs began. " My 
 " Lord I have undertaken this long journey purpofely to fee your per- 
 " fon, and to know by what engine of wit or ingenuity yovi came firft 
 *' to think of this moft excellent help unto Aftronomy, vi-z. the Loga- 
 " rithms ; but My Lord, being by you found out, I wonder nobody 
 " elfe found it out before, when now being known it appears fo eafy". 
 He was nobly entertained by the Lord Napier, and every fummer after 
 that dui'ing the Laird's being alive, this venerable man Mr Briggs went 
 purpofely to Scotland to vifit liim." 
 
 There is a palTage in the life of Tycho Brahe by GafTendi*, which 
 may miflead an inattentive reader to fuppofe that Napier's method had 
 been explored by Herwart at Hoenburg, 'tis in Gaflendi's obfervations 
 on a letter from Tycho to Herwart, of the laft day of Auguft 1599. 
 Dixit Hervartus nihil morari fe folvendi cujtifquem trianguli difficultatem ; fo- 
 lerefc enim multiplicationum^ ac divifionum vice addittones folum, fubtra&iones 
 93 ufurparc ( quod ut Jicri pojfet ^ docuit pojlmodum fuo Logarithmorum Carione 
 Neperus.) But Herwart here alludes to his work afterwards publiflicd 
 
 £ in 
 
 • Tychonij Brahii Vita. Parifius 410, 1654. p. 191.
 
 i8 LIFEofNAPIER. 
 
 in the year 1610, which folvcs triangles by Profthaphan-efis, a mode to- 
 tally difiPercnt from that of the Logarithms. 
 
 Kepler dedicated his Ephcmerides to Napier, which were publifhed 
 in the year 1 6 1 7 * ; and it appears from many paflages in liis letter about 
 this time, that he held Napier to be the greatell man of his age, in tlie 
 particular department to wliich he apphed his abilities : and indeed if 
 we confider, that Napier's difcovery was not, like thofe of Kepler or of 
 Newton, connedted with any analogies or coincidences, which might 
 have led him to it, but the fruit of unaflifted reafon and fcience, we 
 fhall be vindicated ill placing him in one of the higheft niches in the 
 Temple of Fame. 
 
 Kepler had made many unfuccefsful attempts to difcover his canon 
 for the periodic motions of the planets and hit upon it at laft, as he 
 himfelf candidly owns, on the 15th of May, i6i8j and Newton ap- 
 plied the palpable tendency of heavy bodies to- the earth to the fyftem 
 of the univerfe in general ; but Napier fought out his admirable rules, 
 by a flow fcientific progrefs, arifing from the gradual revolution of 
 truth. 
 
 The laft literary exertion of this eminent perfon, was the publication 
 of his Rabdology and Prompmary, in the year 161 7, Avhich he dedi- 
 cated to the Chancellor Seton, and foon after died at Merchifton, on the 
 3d of April, O. S. of the fame year, in the 68th year of his age, and, 
 as I fuppofe, in the 23d of his happy invention. 
 
 In 
 
 • Kepler's Ephemcridcs novse motuum cselcftium ab anno 1617.
 
 / 
 / 
 
 L I F E o F N A P I E R. 19 
 
 In his /^^°'^» ^^'^ portraits * I have feen reprefent him of a grave 
 and f\7<^ countenance, not unlike his eminent contemporary Mon- 
 fiemv^ Peirefc. 
 
 / In his family he feems to have been uncommonly fortunate, for his 
 eldefl fon became learned and eminent even in his father's lifetime, his 
 third a pupil of his own in Mathematics, to whom he left the care of 
 publifliing his Pofthumovis works ; and lofing none of his children by 
 death, he loft all his daughters by honourable or refpedlable marriages. 
 
 He was twice married. By his firft wife, Margaret, the daughter of 
 Sir James Stirling of Kier, defcended of one of the oldeft and moft re- 
 fpedlable gentlemen's families in Scotland, he had an only child, Ar- 
 chibald, his fucceffor in his eftatcs, of whom I fhall hereafter give fome 
 account. By his fecond marriage with Agnes, the datighter of Sir James 
 Chifliolm, of Crombie, he had five fons : John, Laird of Eafter Tonie ; 
 Robert f, who publifhed his father's works, whom I have already men- 
 tioned, the anceftor of the Napiers of Kilkroigh in Stirling fliire ; 
 Alexander Napier of Gillets, Efq ; William Napier of Ardmore ; and 
 Adam, of whom the Napiers of Blackftone and Craigannet in Stirling 
 ihire are defcended. His daughters wer,c, Margaret, the wife of Sir 
 James Stuart of Roflayth ; Jane, married to James Plamilton, Laird of 
 Kilbrachmont in Fife ; Elizabeth, to William Cuninghame of Craig- 
 ends ; Agnes, to George Drummond of Baloch ; and Helen, to The 
 
 Reverend 
 
 • In the Univerfjty Library at Edinburgh, another in the ponciTion of the Lord Napier, and aa 
 8vo print, engraved by Delaram, where he is rcprefented calculating with his bones. 
 t Robert wrote a Chemical Treatifc, ftill prefcrvtd in the family of Napier.
 
 20 L I F E o F N A P I E R. 
 
 Reverend Mr Mathew Bufbane, Redlor of the Parifli o. £i-flj_inc m 
 Renfrew fbirc. 
 
 He was interred in the Cathedral Church of St Giles, at Edinbu-gU, 
 on the eaft fide of its nortliern entrance, where there is now a Stoi^ 
 Tablet, indicating, by a Latin Infcription, that the burial place of the 
 Napiers, is in that place ; but no Tomb has ever been eredled to ike 
 memory of fo celebrated a man, nor can it be required to preferve 
 his memory, fince the aftronomer, geographer, navigator and political 
 arithmetician, muft feel themfelves every day indebted to his inven- 
 tions, and thus a monument is eredled to the illuftrious Napier, which 
 cannot be obliterated by time, or depretiated by the ingenuity of others 
 in the fame department. 
 
 I proceed now to evince more fully the merit of Napier, by giving 
 an account of the flate in which he found Arithmetic, and of the be- 
 nefit it received from his difcoveries. 
 
 SECTION
 
 SECTION 
 
 CONCERNING ARITHMETIC. 
 
 An cum Slatuas et Imagines, non animarumfimulacrafed corporum, Jiudiofi multi fummi homines reliqturiint ; 
 eonjiliorum relinquere, ac virtutum nojlrarum effgiem nonne multo malU debemus, fummis 'mgeniis exprejfam. 
 et polUam ? 
 
 CiCERONIS OrATIO PRO Ap.CHIA POETA. CAP. III. 
 
 Arithmetic is fo necelTary to man, tliat it muft have made its 
 appearance on the firft and rudefl flage of fociety. 
 
 Signs to exprefs numbers were probably in ufe, as foon as figns to 
 cxprefs other ideas. 
 
 The ligns the mofl obvious, and we may venture to fay the firll in 
 ufe, were the fingers. The number of thefe accounts for tlie genera! 
 adoption of numeration by tens. The firft ten numbers have the ap- 
 pellation of digits or fingers, in moft of the languages. 
 
 The next improvement of Arithmetic, feems to have been the ufe of 
 fmall pebbles, or of knotted firings. The words y^ifnom and calculus 
 
 F fignify
 
 12 L I F E, W R I T I N G S, A N D 
 
 fignify both a pebble and an arithmetical operation. The Ruffians, to 
 this day, perform their calculations by means of flringed beads, with 
 great cxatflnefs and expedition. The Greeks and Romans reprefented 
 numbers by the letters of the Alphabet varioufly combined. By means 
 of their notation, the operations of addition and fubtratSlion of integers 
 at lead, were eafily enough performed. But multiplication, divifion, 
 and die extra6tion of roots, were difficult and tedious operations. They 
 muft have efFe<5led them, in a great meafure, by dint of thought. 
 Boethius, who flourilhed towards the end of the fourth century, fays 
 indeed, that fome of the Pythagoreans had invented, and ufed in their 
 calculations, nine apices or charaders, refembling thofe we now em- 
 ploy ; by which thefe latter operations muft have been much fmiplificd.. 
 Thefe figures were known only to a few myfterious men, and it is by 
 no means probable that they were the inventors of them. It is probable 
 that Pythagoras, or fome of his difciples, borrowed them, as they did 
 many other inventions, from the Indians. The merit of the Greek 
 Philofophers, of which Euclid claims a diftinguifhed fhare, confifted in 
 railing Arithmetic, from being a fimple art, to the rank of the fciences. 
 
 Gerbert of Aquitaine, in France, afterwards Pope Sylvefter the fe- 
 cond, having imbibed tlie elements of the fciences, found that the chri- 
 ftian world, at tliat time involved in darknefs, could not furnifh him 
 with fufficient helps for making any great progrefs in them. This in- 
 duced him to fly from the Convent of Fleury, where he had lived from 
 his infancy, to Spain ; where, under the tuition of the Moors, he be- 
 came fo intimately acquainted with the mathematics, that he is faid to 
 have foon furpaffed his mafters. Upon his return to his native coun- 
 try,
 
 INVENTIONS OF NAPIER. 23 
 
 try, about the year 960, or 970, he introduced the ten chara(flers, 
 which form the bafis of our modern Arithmetic* Thefe had been fa- 
 mihar to the Arabs, time out of mind, and the invention of them is, 
 by their writers, afcribed to the Indians *. 
 
 About five hundred years afterwards, our Arithmetic received a 
 mofl: important improvement,- by the invention of decimal fradlions. 
 
 As the invention of thefe fradlions, and of the Logarithms, with other 
 arithmetical improvements, was occafioned by the efforts of ingenious 
 men, to perfedl Trigonometry, it will be proper to give fome account 
 of the rife and progrefs of this mod ufeful branch of the mathematics. 
 
 Trigonometry, confidered as a fimple art, muft have begun with 
 the divifion of lands in every country; but coniidered as a fcience, or 
 as the application of Arithmetic to Geometry, it feems to have had 
 its rife among the hands of the great Hipparchus, about one hundred 
 and forty years before the chriflian asra. Hipparchus was tlie firlt who 
 made ufe of the longitudes and latitudes, for determining the pofition 
 of places, on the furface of the earth. Theon cites a treatife of his, in 
 twelve books, on the chords of circular arcs, which muft have been a 
 treatife on Trigonometry, and is the firft of which hiflory gives any ac- 
 count. Menelaxis, about the end of the firft century, wrote a treatife, 
 in fix books, on the chords ; and there are extant of his three books on 
 Spherical Trigonometry, where that fubjedl is treated in a manner ve- 
 ry profound and cxtcnfive. 
 
 The 
 
 • Wallis,. Montuda, (jfc.
 
 24 L 1 F E, W R I T I N G S, A N D 
 
 The diflicultlcs to be encountered in the folution of triangles, which 
 is the objecl of Trigonometry, regard the tables of the" parts of the 
 circle, the form of the problems to be ufed, and tlie application of 
 thefe problems to pradlice. 
 
 The Ancients, before Ptolemy's time, do not feem to have agreed 
 upon a particular divifion of the radius of the circle *. That indefatig- 
 able Aflronomer, who flouriihed about the year 200, having fimplified 
 the theory of Menelaus, divided the radius into fixty equal parts, and 
 computed on this foundation, the length of all the chords in the femi- 
 circle, correfponding to every thirty minutes. This fexagenary divi- 
 fion, which continued in ufe for many centuries, obliged geometers to 
 make ufe of numbers compofed of integers and fradtions, which occa- 
 fioned much labour and much lofs of time. The table of chords led 
 them to problems very complicated and of difficult execution. Every 
 oblique triangle was to be divided into two redlangxilar ones ; and in 
 order to come at a folution, it was necefTary to raife to the fquare, and 
 to extratft the fquare root of many fractional numbers. 
 
 The Arabs, fometime in the eleventh century, greatly fimplified the 
 theory of Trigonometry, by fubftituting, for the chords of the double 
 arcs, the halves of thefe chords. Thefe lines have been called finus, 
 probably from S. Ins. an abbreviation of the Latin words Jemijes infcrip- 
 tarum\. This improvement paved the way to more fixnple theorems, 
 of which we fhall have occafion afterwards to fpeak. 
 
 About 
 
 • Hanfchii Prcf. in Kepi. Epift. t Montucla Hiftoirc Mathcmatique.
 
 INVENTIONS OF NAPIER. 25 
 
 About the middle of the fifteenth century, George, furnamed Peur- 
 bach, from a village on the confines of Auftria and Bavaria, where he 
 was born, either adopted the finus from the Arabs, or invented them 
 himfelf. He alfo banilhed from Trigonometry the ufc of the fexage- 
 nary calculus, by fuppofing the radius to confifl of 600 000 equal parts, 
 and computing on this foundation the length of the fines correfpon- 
 ding to every ten minutes of the Quadrant. 
 
 John Muller (commonly known by the name of Regiomontanus 
 from the place of his birth, Konigfberg a town in Franconia) the dil- 
 ciple of Peurbach, improved his mafter's idea by maJcing the radius 
 equal to unity or 1,0000000. On this new plan he calculated, with 
 great labour and accuracy, a table of tlie fines for all the minutes of the 
 Quadrant. He alfo was the firft who introduced the ufe of the tangents 
 in Trigonometry ; of which Erafmus Reinoldus of Salfeldt firft con- 
 llrudled a table. To thefe tables Rheticus * afterwards added that of 
 the fecants, which had been invented by F. Maurolyeus of Meffina. 
 
 By means of thefe new tables the art of Trigonometry was not only 
 rendered more accurate than formerly, but one multiphcation or divi- 
 fion was fuperfeded in every geometrical proportion where the radius 
 made one of the terms. The multiplication or divifion, however, of 
 fuch large numbers required much expence of time, labour and atten- 
 tion. 
 
 G Raymarus 
 
 • George Joachim, fo called from his native country. Tliefc appellatives, fo much ufcil after the 
 revival of letters, make it often difficult to difcovcr the real name* of learned men.
 
 26 LIFE, WRITINGS, a :.' d 
 
 R.AYMARUS Urfus, towards the end of the fixteentli centuiy, having 
 cither learned from his preceptor Juflus Byrgius, or difcovered fome 
 new properties of the fines, fliewed, in his Fundamcntum AJlronomkum 
 publillied in tlie year 1588, how thefe might be employed to great ad- 
 vantage in the folution of fome trigonometrical queftions. By lus me- 
 thod, which he calls Pro/lbapha-jxfis, from T^oirhirtcr addilio and 0.(^0.1 ^efif 
 ablatio, the fourth term of a geometrical proportion, having for its firft 
 term the radius equal to unity, may be found by addition and fub- 
 tradlion only; inftead, for example, of multiplying the fine a by the 
 fine of b in the geometrical proportion i : fin. a : : fin. b : fin. c, the fine 
 of c may be had, with much lefs trouble, by fubtrafling half the cofine 
 of the fum of a and b from half the cofine of their difference ; becaufe, 
 as is eafily demonfb'ated, fin. ^x fin« b — l cofin [a — b) — i cofm {a-\-h). 
 
 It was only to a few cafes, however, that the proflhaphasrefis of 
 Raymar could be applied, and the improvements made upon it, by 
 Clavius * Magini j" and others, required fo many precautions that they 
 were not of very great fervice. % ^^^ inconfiderable as thefe abbrevi- 
 ations of calculus were, they were generally ufed by the moft eminent 
 matlaematicians and aftronomers at the end of the fixteenth and begin- 
 ning of the feventeenth century §. 
 
 Such 
 
 * Clavius de Aftrolabio, book i. lemma 53. 
 
 ■)• Magini primura mobili, lib. I. theor. 33. and lib. 11. cap. 2. 
 
 X Qiiod vero Profthaphsrefcin tabulus attinet, fcito me totum hunc annum qua parte et a morbis 
 tX a curis fui vacuus in unlus martis profthaphserefibus excentri vcrfari, nee pudet dicere mc fcopuim 
 nondum attigcfle. Kepler Epift. p. 1 7 1 . 
 
 J Profthaphiretical tables were publifhed by J. G. Herwart, iaiGio.
 
 INVENTIONS OF NAPIER. 27 
 
 Such was the ftate of arithmetical computation, at the time of the in- 
 rentionof the Logarithms, which, as Napier himfelf fays, Omnem illam 
 prijlina; mat he/cos d'lfficultatem perntas e medio tollit ; et ad fubkvandam mcmo- 
 ri(S imbecillitatem itafc accomodate ut illius adminictilo facile ftt, plures quajli- 
 oiies mathematkas unlus horafpatio^ quam prijl'tn'ia ct communiter rccepta forma 
 ftnuum^ tangentimn et fecatitium^ vel integro die abfolvere*. But before we 
 proceed to this mod important difcovery, we fliall give an account of 
 thofe ingenious contrivances, intended to anfwer the fame purpofe, 
 wliich previoully occurred to Napier. 
 
 SECTION II. 
 
 napier's bones. 
 
 1 HE firft of thefe mechanical devices is what our author calls Rabdo- 
 logia, or the art of computing by figured rods. Thefe rods are fquare 
 parallelepipeds three inches in length, and three tenths in breadth. 
 Each of the faces of thefe parallelepipeds is divided into ten equal parts, 
 of which nine are fquares and in the middle, and half of the tenth at 
 one extremity or the top, and half at the other extremity or the bot- 
 tom. Every one of thefe fquares is cut by a diagonal from left to 
 right upwards. At the top of each face is fome one of the ten di- 
 gits o, 1,2, 3, ^c. 
 
 In 
 
 * Log. Canon, dcfcriptio. in dcdic.
 
 23 LIFE, WRITINGS, and 
 
 In the firft fquare below that digit is repeated, in the fecond is its 
 double, in the third it's triple, and Co on. Of thefe multiples of the 
 digit, the figure of units is below, and the figure of tens above the 
 diagonal. The meaning of what has been jull faid will be evident by 
 a little attention to Fig. I. where the four faces of each rod of the 
 fct, recommended by Napier, are unfolded. By means of thefe rods the 
 operations "of multiplication and divifion are performed by addition 
 and fubtradlion. 
 
 The rule for multiplication is — Bring tlie rods to form the multipli- 
 cand at the top of their upper face. Join a rod, having unity at the 
 top of its upper face, to the right or left hand fide ; in which feek the 
 right hand figure of the multiplicator, and write out the numbers cor- 
 refponding thereto in the fquare of thfe other faces, by .adding the fe- 
 veral numbers occurring in the fame rhomboid. Seek the fecond figure 
 of the multiplicator and proceed in the fame manner : arrange and add 
 the numbers wrote out, as in common multiplication ; the fum is the 
 produdl required. To multiply 1785 by 364, for example, I difpofe the 
 proper rods as in Fig. II. ; next to 4 (the firfl right hand figure of the 
 multiplicator) I find o; in the contiguous rhomboid 2 and 2, which ad- 
 ded together make 4; in the next 3 and 8 which make i and a furplus 
 of ten ; and in the laft 2 and 4 which, together with unity for the ten I 
 had in the former rhomboid, make 7. Thefe numbers o, 4, i , 7, I fet 
 down one after the other as I find them, proceeding from right to left. 
 I go on in the fame manner with 6 and 3 (the other figures of the mul- 
 tiplicator) ; and, after arranging and adding the partial producls I find 
 tlie total produd required. Thus, 
 
 .364
 
 INVENTIONS OF NAPIER, 29 
 
 364 
 
 7140 
 10710 
 5355 
 
 649740 
 Tbe rule for div'ifion. Bring the rods to form the divifor at the top 
 of their upper face. Join a rod having unity at the top of its upper 
 face, to the right or left hand fide. Defcend under the divifor till you 
 meet thofe figures of the dividend wherein it is firft required how oft- 
 en the divifor is found, or the next lefs number, which fubtra(5l from 
 the firft figures of the dividend, and put for the firft figure of the quo- 
 tient the correfponding number on the fide face. Bring down, one af- 
 ter the other, the remaining figxires of your dividend as in common di- 
 vifion, and proceed in the fame manner till you have finiflied the ope- 
 ration. Let it, for example, be required to divide 649740 by 364. I dif- 
 pofe the rods as in Fig. III. The next lefs number vinder the divifor 
 364 to 649 (the firft figures of the dividend) I find to be 364 itfelf 
 which I fubtradl from 649 putting i, the number correfponding on 
 the fide face, for the firft figure of my quotient : to my remainder 285 
 I bring down 7 the next figure of my dividend. The next lefs num- 
 ber to 2857 "i^der the divifor I find to be 2548, which I fubtradl from 
 2857, putting 7, the number correfponding in the fide face, for the fe- 
 cond figure of the quotient. I go on in the fame manner till I have 
 brought down the other figures of the dividend and completed my 
 quotient as follows. 
 
 H 649740
 
 30 LIFE, WRITINGS, and 
 
 649740(1785 
 364 
 
 2857 
 2548 
 
 3094 
 2912 
 
 1820 
 1820 
 
 (o) 
 
 Although the extradlion of the fquare and cube roots may be pret- 
 ty expeditioufly performed by the rods, Napier propofes an auxiliary 
 lamella for the abridgement of it. It would ferve little purpofe to give 
 a particular defcription of the lamella, or an account of the manner of 
 ufing it. Its length and thicknefs are the fame with thofe of the rods, 
 and its breadtli quadruple. Its two faces are divided and marked as in 
 Fig. IV. To find out the way of operating by it will be no difficulty 
 to any body who is a httle acquainted with ai-ithmetic and has time 
 to Ipare. 
 
 Another of Napier's contrivances is his maltiplicationis promptu- 
 arium. 
 
 This machine confifts of a box of figured lamellje. The lamellae, two 
 hundred in number, are each eleven inches in length and one inch in 
 breadth. Each of thefe lamellae is divided into eleven equal parts of 
 which ten in the middle are fquares, and two thirds of the eleventh at 
 
 one
 
 INVENTIONS OF NAPIER. 31 
 
 one extremity, and one third at the other. Every one of thefe fquares 
 is divided into nine lefs fquares, one hundred of the lamellae are each 
 one fourth of an inch in thicknefs, and the other hundred one eighth. 
 Suppofe the former, which we fliall call diredl lamellae, placed fo that the 
 greater margin may appear at top and the lefs at bottom ; and the latter 
 which we fhall call tranfverfe, placed laterally, with the greater margin 
 to the right and tlie lefs to the left hand. In this pofition every fquare 
 appears cut by a diagonal (faint in the fmall but ftrong in the great 
 ones) from the left to right upwards. Each of every ten both of the 
 direcl and of the tranfverfe lamellae has fome one of the ten digits 
 o, I, 2, 3, ^c. infcribed on its greater margin. The multiples of the 
 digit on the margin of a dire(5l lamella are difpofed in each of its 
 greater fquares as pointed out by Fig. V. where a reprefents the digit 
 itfclf, Z" the right hand figure, and ^' the left hand figure of its double ; 
 c and c' the right and left hand figures of its triple (the plain letters 
 being above and the accented ones below the diagonal of the figure) ; d 
 and d' thofe of its quadruple, and fo on. In tlie tranfverfe lamellae 
 thofe which have on the margin are untouched ; thofe which have 
 unity on the ipargin have the loculus correfponding to a cut out; tliofe 
 which have two on the margin have the loculi of 6 and 6' cut out ; 
 thofe which have 3 the loculi of c and c' ; tliofe which have 4 the lo- 
 culi of i/ and d\ "Ujc. This will be fufficiently evident by infpedling 
 Fig. VI. where it is examplified in a dire<fl lamella titled with the digit 
 4, and in a tranfverfe one with 7. The box fitted to receive thefe la- 
 mellae is of a cubical form ; fomething more tlian eleven inches wide 
 and nearly eight inches high ; fee Fig, VII. Two of its contiguous 
 fides, which we fliall dillinguifli by the names of left and right, are 
 
 divided
 
 . : LIFE, WRITINGS, and 
 
 divkled into twenty parts, each equal in length to the breadth of ten 
 lamclLx, and in height to the thicknefs of a dircdl and of a tranfverfe 
 lamella alternately. The greater divifions on the left fide are cut out, 
 and tile lefs on the right fide. Into the box through each of the for- 
 mer, with their tided ends foremoft, ten direct lamellae of the fame 
 title are inferted with their untitled ends foremoft, and an equal num- 
 ber of the tranfverfe ones of the fame title, through each of tlic latter. 
 Thofe titled o are in the uppermoft divifions, and tliofc titled i, 2, 3, 
 s^r. in the refpe(!^ive divifions below. 
 
 b 
 
 , MidtipUcatiori by the prompttiary is performed as folloirs. The fir ft, or 
 right hand, fecond, tliird, 8cc. figure of the multiplicand is exhibited 
 by the title of a lamella taken from the firft, or right hand, fecond, 
 third, &c. column of the left fide of the box and placed on its lid 
 exa(5Uy above and as it lay in that column. The empty fpace, if any, 
 towards the left is to be covered with blank lamellae. The firft, or 
 right hand, fecond, third, 8<:c. figure of the multiplicator is exhibited 
 by the title of a lamella taken from the firft, or left hand, fecond, 
 third, &c. column of the right fide of the box and placed on the for- 
 mer lamella exa6lly above, and as it lay in that column. The remain- 
 ing fpaces, if any, towards the right are to be covered with blank la- 
 mellae. All the ufeful multiples on the dire<5l lamellae appear through 
 x\\tfetieJlellcEy and all the ufelefs multiples are hid. All the numbers be- 
 ginning at the corner next the firft or right hand figures of the multi- 
 plicand and multiplicator, lying between the united ftroug diagonals, are 
 to be added feverally ; the right hand figures of the fums, miarked down ; 
 and I for every i o, carried to the next place, till we come to the oppo- 
 
 fite
 
 INVENTIONS OF NAPIER. 33 
 
 fite comer : and the work is done. This operation, we trull, is defcri- 
 bed with fuiEcient accuracy and plainnefs to fuperfede the ncceflity of 
 an example. In order that diviiion may be performed by the Promptu- 
 ary, it muft firfl be converted into multipHcation by means of tables 
 drefled on purpofe, or of tables of the fines, tangents and fecants, coa- 
 ftru(5lcd on the hypothefis of the radius being equal to unity, followed 
 by a certain number of Zeros. That this may be accomplilhed by 
 thefe lafl:, look for the co-fecant, or co-tangent of the arc which has 
 tlie divifor for its fine or tangent. Make the co-fecant or co-tangent 
 found tlie multiplicand, and the dividend the multiplicator ; or con- 
 verfely. Find the produ(5l by the promptuary as above diredled. This 
 producSl, a number of the right hand figures according to the number 
 of zeros in the fquare of the radius being marked off as decimals, is 
 the quotient required. The reafon of this is obvious : the co-fecants or 
 co-tangents being third proportionals to the fines or tangents and the 
 radius or unity ; to multiply any number by one of the two firfl, or to 
 divide it by the correfponding one of tlie two fecond of thefe lines, is 
 one and the fame tlfing. 
 
 LOCAL
 
 34 LIFE, WRITINGS, and 
 
 LOCAL ARITHMETIC. 
 
 LOCAL Arkhmotie, another ingenious invention of Napier, i» 
 die art of calculating by means of counters properly placed on a chefs 
 board, or fiinilar table. Two contiguous margins (which we fliall di- 
 ftinguifli by the names of left or inferior, and I'ight or lateral) of tliat 
 table, are divided into a number of parts equal to that of their adjoin- 
 ing fquares. The inferior divifions beginning at the right and the la- 
 teral at the left have fuccelTively infcribed in them the fucceilive terms 
 of the geometrical progreflion i, 2, 4, 8, 1 6, 32, Isfc. wliich ai^e called 
 local numbers. 
 
 Common numbers are reduced to local by fubtracflion, and local 
 numbers to common by addition. The common number 1 8 j^^ for ex- 
 ample, exprefled in local numbers v^ill be found to be 1024 ; 5 1 2 ; 256 ;. 
 64 ; 16; 2 and i : and vice verfa. The firft of thefe redu(5lions is ne- 
 cefTary before, and the fecond after any of the operations of common 
 arithmetic are performed by this contrivance. By the help of a very 
 fimple table, reduction may be performed with eafe and expedition. 
 
 To AdiL Put a counter for each local number in the proper .place 
 on the lateral or on the inferior margin of your table. For every two 
 counters found in the fame place, put one in the next higher, after re- 
 moving them. Repeat tliis till no place fliall contain more than one 
 counter. The counters left indicate the numbers required. Thus let 
 it be required to find the fum 1875 ; 258, and 1099. I put the coun- 
 ters
 
 INVENTIONS OF NAPIER. ^^ 
 
 ters at 1024; 512; 256; 64; 16; 2 and i, tlie local numbers of die 
 firfl ; at 256, and 2, thofe of the fecond ; and at 1024 ; 64, 8, 2, and i, 
 thofe of the third. At i. — I find two counters which I remove, and 
 put a counter at 2 wliere I find other three. I tiike up thefe four and 
 put two, in the next place 4 &c. and proceeding in this manner I find 
 at laft a counter at each of the following numbers 2048 ; 1024 ; 128, 
 and 32, which form 3232 the fum fought. 
 
 To Subtract. Put a counter for each local number of the greater of 
 tlic two quantities, at its proper place, a little to one fide,[.on the infe- 
 rior margin ; and one for each of the local numbers of the lefs of the 
 two quantities, at its proper place, a little to the other fide, on the fame, 
 margin. Remove the counters found on oppofite fides of the fame place. 
 Change the fide of the lowefl counter remaining ; take oflf that above 
 it ; and put a counter in each place bet^veen them. Remove as before. 
 Repeat this till tliere Ihall be no counters on one of the fides of the mar- 
 gin ; and thofe on the other will indicate the remainder. Let it be pro- 
 pofed, for example, to fubti'adl 1099 from 1875. I put counters at 
 1024; 64; 8 ; 3 and i, to the left of the lateral margin, and at 1024; 
 512; 256; 64; 16; 2 and I, to the fiit of that margin. Finding a 
 counter on each fide of the numbers 1024; 64; 2, and i, I remove 
 them. My lowefl counter is to the left of 8. I put it to the right and 
 take up 16. above it ; as there are no intermediate places, and as the re- 
 maining counters ai'e on the fiime fide of the margin my operation is fi- 
 niflied. The remainder is 512; 256, and 8j or 776. 
 
 Multiplication,
 
 2,6 LIFE, WRITINGS, and 
 
 Multiplication, Divifion, and the cxtradlion of die fquarc root, 
 may alfo be performed on the margin : but they are performed with 
 much greater eafe and clearnefs on two contiguous margins and the 
 fquares of the table. On thefe laft the counters have two different 
 movements ; the one parallel to the fides like that of the tour, and the 
 other diagonal like that of the bilhop, on the chefs board. Every 
 fquare of the table is faid to have for its value one of the equal num- 
 bers (on the two margins) between which it lies diagonally. The two 
 fides of a fquare formed by counters in the area of the table, parallel 
 to the inferior and lateral margins, we fliall call a Gnomon : this gno- 
 mon confining of 3, 5, 7, 8cc. counters is faid to be congruous when 
 its value can be fubtracled from the numbers left marked upon the 
 margin. 
 
 To Multiply. Mark witli counters the local multiplicator in the 
 inferior and the local multiplicator in the lateral margin. From the 
 middle of the marked places let points be fuppojed to move perpendi- 
 cularly into the table, and put a counter at each interfe^lion. Remove 
 the counters on the margins. Bring the counters in the fquares of the 
 tables to their values in one of the margins ; add, if neceflary, and the 
 work is done. Suppofe, for example, 19 is to be multiplied by 13. I 
 mark with coiinters Fig. VIII. the numbers 1,2, and 16, on the infe- 
 rior and the numbers i, 4, and 8, on the lateral margin, having pla- 
 ced other counters retSlangularly in the table, I remove the marginal 
 ones. Thofe other counters I bring up, one by one, to their proper pla- 
 ces in the lateral margin ; and, after adding, I find a counter at each 
 
 of
 
 INVENTIONS OF NAPIER. 57 
 
 of the following numbers, 12S ; 64; 32 ; 16; 4 ; 2, anl i, which fo:m 
 my produdl, 247. 
 
 To Divide. Mark with counters the local dividend in the latca', 
 and the local divifor in the inferior margin, beginning at the fquare 
 where a point, defcending diagonally from the angle above the highcfl 
 number of the dividend, would interfedt a point afcending perpendi- 
 cularly from the higheft number of the divifor ; place a feries of coun- 
 ters parallel to the diviior : If this feries is equal or iaferior in value 
 to the higher number of the dividend fubcraft it from them ; and if 
 otherwife, bring it down one, two, &c. Heps and fubtradl. Repeat 
 the operation till either nothing, or at leaft a numberlefs tlian the divi- 
 for, fhall remain on the lateral margin. Thefc fcriefes point to the 
 numbers that form the quotient. For example let it be required to di- 
 vide 250 by 13. I mark, Fig. IX. the numbers 128 ; 64; 32 ; 16 j 8 j 
 and 2, in the lateral, and 8 ; 4, and i in the inferior margin. 
 
 My firfl feries points to i5. I fubtra(a it from the dividend and fmd 
 remaiiiijQg 32 j 8, and 2. 
 
 My next feries pointing to 4 is too great to be fubtradlcd, for wliich 
 reafon I bring it a Hep farther down. 
 
 After fubtra(5^ing, there remains 16. In the fame manner my third 
 feries pointing to 2 I mufl bring to point to i ; which fabLradled, there 
 remains counters on the dividend at 2 and i. My quotient is there- 
 fore 16; 2, and I, or 19 ; and 2 and i, or 3 over. 
 
 K r#
 
 33 L I F E, \V R I T I N G S, A N D 
 
 To extras the fqiiare root. Mark the number locally in tne lateral 
 margin. From the angle formed by the meeting of the inferior line 
 with the lateral, let a point afcend diagonally till it arrive in a fquare 
 of the fame value with the higheft number that can be fubtradled from 
 the number whofe fquare root is fought. In this fquare place 4 
 counter, and fubtracl its value from the number marked in the margin. 
 Form the congruous gnomons, which from the forefaid fquare have each 
 tlitlr upper counter in a line perpendicular and their left hand inferior 
 one in a line parallel to the lateral margin : and liibtradl their value one 
 by one from the marked remainders. The counters, lying perpendi- 
 cularly to either of the margins, point out the fquare root. Let it be 
 propofed, for example, to fmd the fquare root of 2209; I mark the num- 
 bers 2048; 128, 32, and I on the lateral margin. Fig. X. Subtracting the 
 value 1 024 of the firft counter placed in the table as directed, the re- 
 mainders ^re 1024, 128, 32 and i. From thefe taking the value 512 
 and 64 of the firfl congruous gnomon, there remain 512, 64, 32 and i. 
 From thefe taking the value of the fecond congruous gnomon 256, 
 64 and 16, there remain 64, 16, 8, 4 and i : and from thefe taking the 
 value of the fourth congruous gnomon, 64, 16, 8, 4 and i, there re- 
 mains nothing. The fquare root, as indicated by the diredion of the 
 coiinters in the table, is 32, 8, 4, 2 and i, or 47. 
 
 What is above faid will, it is hoped, be fufficient to give a clear idea 
 of the form and ufe of thofe of Napier's arithmetical inftruments, which 
 feemed to him worthy of being communicated to the public. The rea- 
 fons on which the different operations are founded, depending upon 
 the coniliM^lion of the macliines and the obvious properties of num- 
 bers,
 
 I N V E N T I O N S o F NAPIER. j^ 
 
 bers, mufl occur to every reader endowed with a moderate iliait; of 
 attention. The hint of the rods, or vlrgulcs numeratriccs and of the 
 promptuaty which is only an improvement of the rods, ieems to have 
 been taken from the Abacus Fythagoricus or common multiplication tableJ 
 Napier's acquaintance with chefs, the mod ingenious of all games, and 
 at which mathematicians are commonly the befl players, occaHoned his 
 difcovery of the Arlthmetica localis. The Promptuary, at leafl: for multi- 
 plication, is greatly preferable to the rods and the chefs board ; for the 
 partial produ(5ls of two numbers, confiRing of even ten Figures each, 
 may, by a little pradlice, be exhibited on that machine in the fpacc of 
 a minute, and no numbers require to be written out, excepting the 
 total produdl. Had the logarithms remained undifcovered, the promp- 
 tuary, in all probability, would have become univerfally familiar to thofe 
 who were engaged in tedious calculations. But to thofe who are ac- 
 quainted with the logarithms, Napier's arithmetical machines and thofe 
 afterwards invented, a few of wliich we fliall enumerate, although the 
 montiments of genius, mud, in general, be regarded as mathematical 
 curiofities of no ufe. 
 
 Perhaps put into the hands of young people learning arithmetiCj 
 they might make them fond of that Iludy. 
 
 Shickartus in a letter to Kepler, written in the year 1623, informs 
 liim that he had lately conilru<!!led a machine confiifting of eleven en- 
 tire and fix mutilated little wheels, by which he performed the fouT 
 arithmetical operations*, Pafcal, in the year 1642, at the age of nine- 
 teen* 
 
 • Kcpl. Epift. p. 683*
 
 4« L I f £, W R I T I N G S, A N » 
 
 teen, invented a machine with which all kinds of computations may 
 be made without the pen, without counters, and without the know- 
 ledge of any rule of arithmetic. I have not been able to meet with any 
 defcription of it. It muft however have been of a very complicated 
 nature as its author was two years in bringing it to perfedlion, owing 
 to the difficulty he found to make the workmen imderftand him tho- 
 roughly *. The French writers agree in calling it admirable ; f but the 
 jiame ot PrJcal perhaps does it more honour than it deferves. This 
 machine is preferved in the cabinet of the king of France and in thofe 
 of a few others %. 
 
 The Marquis of Worcefler, a man of genius but a plagiary, men- 
 tions in his fcantlings of inventions, publifhed in the year 1 6^§, an in- 
 ftrument whereby perfons ignorant in arithmetic may perfedly obferve 
 numerations and fubtra<flions of all fums and fra<5licns §. Whether he 
 here- refers to feme of Napier's inftruments, to Gunter's fcale, of which 
 I Hull afterwards fpcak, or to fome invention of his own is uncertain. 
 
 About the year 1670, **Sir Samuel Moreland contrived two arith- 
 metical inftruments ; one lor addition and fubtradlion, and the other 
 for multipiicaton, diviCon, and the extracflion of the fquare, cube, and 
 fquare cube roots, the defcription of which lie pubhihtd at London, 
 anno 1671 If* 
 
 Much 
 
 • ies bommes illuftres par Perranlt rie de Pafcal. -j- Bayle Chauffcpie, Baillct, Perrauh, &€• 
 % Pref. Penfees dc PafcaL § N° 84. Glag. Edit, 1767. 
 
 ** Moreland's Inftnuncnt of excellent ufe as well at fca as at land, invented and varioufly cxpeti- 
 ■dented in the year 1670, Loud. 1671. FoL 
 ii See alfo Phil. Tranfaa. N^ 94. p. 6045.
 
 INVENTIONS OF NAPIER. 41 
 
 Much about the fame time, Mr George Brown, afterwards Miniflcr 
 of Kilmaures in Scotland, invented a maciiine which, in his account of 
 it publiflied at Edinburgh in the year 1700, he calls the Rotula Aritb- 
 mefica. This machine confiils of a box containing a circular plane move- 
 able on a center pin and fixed ring, whofe circles are defcribed from 
 the fame center. The outermoft circular band of the moveable, and 
 rfie innermoll of the fixed, are each divided into a hundred equal parts, 
 and thcfe parts are numbered o, i, 2, 3, 8cc. Upon the ring there is 
 a fmall circle having its circumference divided into ten equal parts, 
 furniihed with a needle which ihifts one part at every revolution of the 
 moveable. By this fimple inftrument are performed the four arithme- 
 tical operations not only of integers but even of decimals finite and 
 infinite *. 
 
 Some time before Mr Brown's little book appeared, Mr Glover had 
 publiOied a Rotie Arithindique fimilar to the Rotula but not fo perfcdl. 
 It would appear however that that gentleman had got fome hints of the 
 CDnftru(5lion of the Rotula from a brother of his own who had been one 
 of Brown's pujiils in i674f. 
 
 In the year 1 725, an inftrument invented by M. de I'Epine of a more 
 fimple conftruction and eafier in its operations than Pafcal's ; in 1730, 
 another by Mr BoiflTenJeau, by which calculation is performed without 
 writing; and in 1738 a third by Mr Rauflin, confifting of rods diffe- 
 rent from thofe of Napier, were approved of by the French academy %. 
 
 L Sam 
 
 • One of tlu'fe mSchines is in the library Moniriog to the faculty of advocates at Edinburgh, 
 f Rot. Ariilim. Pv.l, \ »S. '.'.ir P.mi memoin for thcfa years. Hilloire.
 
 42 L I F E, \V R I T I N G S, 8cc. 
 
 Sam Reyer invented, at what time I liave not been able to learn, x 
 kind of fexagcnal rods in imitation of Napier's, by wiiicli iexageuary 
 arithmetic is eafily performed*. 
 
 I have an arithmetical machine which came into my poffefhon from 
 my uncle George Lewis Erflcine who, though born deaf, by the aflif- 
 tance of the learned Henry Baker of the Royal Society at London, ac- 
 quired not only the ufe of fpeech and the learned languages but a deep 
 acquaintance with ufeful literature. This machine confifts of a fmall 
 fquare box furniflied with fix cylinders moveable round their axes. 
 Upon each of thefe cylinders, which are only Napier's rods, are engraven- 
 the ten digits, and their multiples. From a perpetual almanac on the 
 out fide of the box, it would appear that this macliine was conllru(5tcd. 
 in the year 1679. 
 
 SECTION. 
 
 *5ee Chaiubei's Didion. Article Arithractit.
 
 SECTION III. 
 
 Napier's theory of the logarithms*: newton's ideas op 
 rluxions, borrowed from napier. 
 
 I Shall now proceed to unfold the Logarithms, the difcovery of which 
 has juilly entitled Napier to the name oi the greateji Mathematician of bis 
 Country. Let two points, the one in N, and the other in L, (Fig. XI.) 
 having at firft a fimilar velocity, move along the indefinite llraight lines 
 C N D and K L A ; die firfl increafing its velocity or diminiihing it 
 according to its diflancc from a fixed point C, and the fecond preferving 
 its velocity without augmentation or diminution. Let the former, in a 
 certain time, arrive at any point N' or n', and the latter in the fame 
 time at the point L' or 1' : the fpace L L' or L 1' defcribed by the fecond 
 moveable point is faid to be the Logarithm of the dillance G N' or C n' 
 of the firfl from the fixed point C. 
 
 I. The Logarithm of CN or unity is zero : for the firfl moveable 
 point not having left N, the fecond has had no time to defcribe any 
 fpace. 
 
 • Tlie term Logarithm was firft ufed by Napier after the publication of the Canon in whidi he 
 nfes tlic terra of nunurui arlijkialij.
 
 44 LIFE, WRITINGS, and 
 
 2. The Logarithms of the terms of a geometrical ferles are in 
 aritlimetical progreflion : for let N N', N'N", N'TSI'", &c. or N n', n'a", 
 n"n"', Sec. be continual proportionals, they will be defcribed by the 
 firft moveable in equal times, and the equal fpaces L L', L'L" L"L'", 
 &c. or L r, ri" I'T", &:c. will be defcribed by the fecond moveable in 
 the fame times. Now it is eafily demonftrated that C N, C N', C N", 
 &c. or C n, C n' C n", Sec. are in ^ometrical progreflion, and it is evi- 
 dent that their refpe<5live logarithms o, L L', L L", &c. or o, L L', 
 2 L L' &c. and o, L T, L 1", Sec. or o, L 1', 2 L 1', Sec. arc in arirh- 
 m^etical progreflion. 
 
 3. The logarithms of quantities lefs than CN are negative, if thofc 
 of quantities greater than C N are pofitive ; and converi'ely : for if C n" 
 C n', C N C N', C N" are continual proportionals, in order that their lo- 
 garithms 2 L r, L r, o, L I/, 2 L L', 8cc. may be in arithmetical pro- 
 greflion it is ncceflary that the terms on different fides of zero fliould 
 have oppofite figns. Hence, 
 
 4. The logarithm of any quantity is the fame with that of its re- 
 ciprocal, the fign excepted. 
 
 5. The number of fyftems of logarithms is infinite: for the ratio 
 of C N to C N' and L L' are indeterminate. 
 
 m 
 
 6. The logarithms of anyone fyflem, are to the correfpondent ones 
 of any otlier, as tl:ie value cf L L' in the firfl fyftem, is to its value in 
 
 the
 
 INVENTIONS OF NAPIER. 45- 
 
 the fecond. From the 2d propofition the four following, expreflcd in 
 the language of arithmetic, are ealaly deduced. 
 
 7. The logarithm of a producfl is equal to the fum of the logarithms 
 of its fadors. Thus the logarithm of GN' X CN" is LI,'+ LL"=LL'": 
 for CN X CN' = CN"'. 
 
 8. The logarithm of a quotient is equal to the difference of the lo- 
 
 CN"' 
 garithms of the divifor and dividend. Thus the logaritlim of y^^, - i« 
 
 ■ CN'" 
 LL"'--LL'=LL": for Xt- = CN". 
 
 CN 
 
 9. The logarithm of the power of a quantity is equal to the pro- 
 du(fl of the logarithm of that quantity by the index of its power. Thus 
 the logarithm of CN'is 3LL'=LL"': for, GN' = CN"'. 
 
 10. The logarithm of the root of a quantity is equal to the quoti- 
 ent of the logarithm of that quantity by the index of its root. Thus 
 the logarithm of yCN"' is ^LL'; for yCN"' = CN'. 
 
 From the 7th and 8th propofitions the two following are evident. 
 
 11. The logarithm of an extreme or mean term of a geometrical 
 proportion, is equal to the diflference of the fum of the logarithms of 
 the means or extremes and the logarithm of the other extreme or mean. 
 
 M J2.
 
 ^(i ' L I F E, W R I T I N G S, A N D 
 
 12. If the logarithms of all the primary numbers are known, tlioib 
 of all the compofite numbers may be found by fimple addition ; and if 
 all the latter are known, all the former may be known by fimple fub- 
 tradion. 
 
 From the 2nd or the 9th and loth propofitions. 
 
 13. The logarithms may be thus defined, Numerorum proporttonalium 
 aquid'ifferentes comiles ; or more properly (as their name, "Koyuv aa<^^oV, im- 
 ports) Numeri rationem exponentes ; becaufe they denote the rank, order, 
 or diflance, with regard to unity, of every number in a feries of con- 
 tinued proportionals of an indefinite nvmaber of terms. 
 
 14. The logarithm LI' of any quantity Cn' is greater than the dif- 
 ference Nn' between CN or unity and that quantity, and lefs than that 
 difference, increafed in the proportion of CN to the faid quantity : for 
 the velocity of. the fecond moveable defcribing LI' being greater than 
 that of the firft defcribing Nn' during the fame time, LI' is greater than 
 Nn' or CN — C; and the velocity with which NN' is defcribed, being 
 greater than that with which LI' is defcribed, in an equal time, LI' is 
 lefs than NN' or CN'—CN or [fince Cn': CN : : CN'], [CN — Cn'jX 
 
 CN 
 
 Cn' 
 
 Hence, 
 
 15. If a quantity Cnl differs infinitely little from CN or xmity, it« 
 
 logarithm LI' will be equal to I Jl -I — l J the arith- 
 
 2Cn' 
 
 metical.
 
 INVENTIONS OF NAPIER. 47 
 
 mctlcal, or to [CN — Cn] x \/CN the geometrical mean between it» 
 
 Cn' 
 Jimits above flated. 
 
 16. The difference I'l" of the logarithms LI' and LI" of any two 
 quantities Cn' and Cn" is lefs than the difference n' n" of thefe quanti- 
 ties increafed in the proportion of the leffer Cn" to CN or unity ; and 
 greater than the faid difference increafed in the proportion of the great- 
 er Cn' to CN or unity : for reafoning in the fame manner as in the 14th 
 propofition I'l" will be found to be lefs than NN' or [fince Cn": CN : : 
 n'n":NN'] CN x n'n". Hence, 
 
 1 7. If the difference of two numbers Cn' and Cn" is infinitely fmall, 
 the difference of their logarithms will be expreffed by the arithmetical 
 (Cn'-l-Cn") X (Cn' — Cn") X CN or the geometrical mean Cn'— Cn" 
 
 2 Cn'X Cn" 
 
 yCn' X Cn" 
 
 X CN between its limits above dated. Beautiful, ingenious and pro- 
 found ! Such is the manner in which Napier conceived the generation of 
 ntmibers and their logarithms, and fuch are fome of their relative pro- 
 perties which naturally flow from it. Thofe who are acquainted with 
 Newton's manner of explaining the doctrine of fluxions, mufl be ft:ruck 
 at its refemblance to this of our Scotifli Geometer. This refemblance, 
 or rather identity, is confplcuous not only in their ideas but in their 
 very words. The explanation of tlie firfl: definition in the Canonis mi- 
 r'lfici defcr'tptio is in the following terms : Sit piin&us A, a' quo ducenda 
 fit linea fluxu altcrlus pmiSl'i^ <i"if't B. Fluat ergo primo moment© B cb A 
 
 in
 
 ^8 L I F E, W R I T I N G S, A N D 
 
 in C. Secundo momento a C in D. I'ertio momento a D in E at que it a de- 
 inceps in iiifinitum defcribendo lineam A CD E F, &cc. hitervallis AC, CD, 
 DE, EF et ceteris deinceps cequalibus^ et mometitis aqitalihtis dtfcriptis^ &c. 
 I the appendix to the Canonis mirijici con/iru&ioy tinder the article Habi~ 
 tudities Lcgarithmorum^ he thus cxprefles the relation betweeh two natu- 
 ral numbers and the velocities of the increments or decrements of their 
 logarithms ; Utjinus major ad minorem ; ita velocitas increiyienti aut dccre- 
 menti apud majorem. What difference is there betwixt this language and- 
 
 that of the great Newton now in ufe x : y : : Log. y : Log. x * ? 
 
 The feeds of the invention of the logarithms were perceived by the 
 ancients as well as by the moderns, upon the revival of fcience in Eu- 
 rope, before the time of Napier. In the elements of Euclid, and in the 
 Arenarius of Archimedes f , thefe great men feem to have been very well 
 
 acquainted 
 
 * See Hutton's Conllniftion of Logarithms, p. 42 and 48. 
 
 ■j- [In the Arenarius of Arcliimcdcs] Without entering in this place, on the repulfion of the rccci»» 
 ■d opinion, that this great Mathematician had made the firft ftcp towards the knowledge of the Lo- 
 garithms, I (hall content myfelf with giving the refult of the enquiry, by one of the ableft Mathe- 
 maticians in the country, to whom I addrefled myfelf, when I firft fet myfelf to produce this work, 
 and who having fuccefsfidly illuftrated the difcoveries of the Prince of Engh'(h Mathematicians, glad- 
 )y came forward to contribute his (hare to the triumph of our Scotifli Newton. 
 
 Archimedes demonftrates a Theorem concerning numbers, made by the mutual multiplication of 
 the terms of a geometrical progre(rion ; by means^f which Theorem the principles of Logarith- 
 mic computation may ea(i]y be demonftrated. Archimedes, therefore, had he been furnifhcd with ta- 
 Ues of Logarithms, woidd have known how to ufe them : But it appears not, tliat lie was poire(red of 
 any principles, which could lead him to tJie formation of Logarithms. He could avail liimfelf, indeedl,. 
 of the indices of the powers of numbers, to abridge the labour of multiplication, as we now avail 
 
 MvfdVes o£ Logarithms for the like purpofe ; But the gu'ph tttween this n.cthod by the Natuiul In- 
 
 diftt
 
 INVENTIONS OF NAIPER. 49 
 
 acquainted witli the correfpondence of an arithmetical to a geometrical 
 
 nrnarpflion *'_ 
 
 progreflion *'. 
 
 Michael Stifelllus, a German Arithmetician, who flourifhed about 
 the middle of the fixteenth century, in his Arithmetica Integra ftated the 
 
 comparifon between the feries •] ' ' ^' ' > 32> (. ^^^ obferving 
 
 that the produtfl or quotient of any two terms of the former correi^ 
 poiadcd to the fum, or difference, of the cquidiftant terms of the latter. 
 
 N Whether 
 
 ^iees, and the method of Logarithms, is wider than it may at firil feem. Any Number, not itfclf 
 arifing from a root, is the root of a difilnit progrcCion of Powers. Ilencc tlicre are as many diflindl 
 progrefTions as there are numbers not aftually powers : And in all thefe progreflions tie homologou* 
 powers have the fame exponents or indices. Thus 3 is the exponent of the number 8 in tl-.e ferie* 
 of the powers of 2. But 3 is equally the exponent of 27 in the feries of the powers of 3 ; of 125 in 
 the feries of the powers of 5 ; of 343 in the feries of the powers of 7 : and univcrfaUy of all cubic nuinf 
 bers ; fo 4 is the exponent of all biquadratic numbers ; 5 of all quadrato-cubic ; and fo on. A num- 
 ler therefore is not fufficiently charafterifed by its exponent unkfs it be known to what feries of pow» 
 ers it belongs, that is from what root it arifes. Add to this that many numbci-s fall into no natural 
 feries of powers. Thk method therefore of computing by the natural indices of powers arifing from 
 the natural numbers as roots, will only ferve the purpofe of rude calculations leading to fome very ge- 
 neral conclufions, and mull fail in all inflances in which accuracy is required. Archimedes never thought 
 of confidering all numbers as exprefllons of proportions, capable of being univL-i fally included in one 
 general feries of ratios, which notion is the true bafis of Napier's great invention, as will be more f.il- 
 ly explained hereafter. For the invention in cfTcft was this ; that he found a method to raife a feries 
 of proportionals, in which all numbers fhoiild be comprifed, in which every number of confcqucncc 
 had its own particular exponent, and to find the ^ponent of any given number, or tlie number of any 
 given exponent in that univerfal feries." 
 
 In the courfc cf this work, it will be fulHcienly proved, tliat Napier was as much the firfl to con- 
 ceive ns to execute this wondcrfiJ projeft. 
 
 • Thofc who wi(h to rccolleft how much we are indebted to the ancients, in tliis as well as in many 
 ether departments of fcicnce?, will read with plcafurc Mr Dutcn's Inquiry into the origin of the ^'f- 
 itovcriiJ altvllattd to the Tiwtknis.
 
 50 L I F E, W R I T I N G S, AND 
 
 Whether Napier ever faw or heard of this remark of Stlfeilitis it 
 not known, nor indeed is it of any confcqiience ; for it cannot fail o£ 
 prefenting itfelf to any perfon of moderate acutenefs who happens to 
 be engaged in arithmetical queftions of this nature where the powers of 
 numbers are concerned. It is not therefore this barren though funda<- 
 mental remark that can entitle him who firfl mentioned it to the name 
 of the inventor of the logarithms. The fupcrior merit of Napier con- 
 fifls in having imagined and afligned a logaritiim to any number what- 
 ever, by fuppofing diat logarithm to be one of the terms of an infinite 
 arithmetical progrcflion, and that number one of the terms of an infi- 
 nite geometrical progreihon whofe confecutive terms differ infinitely 
 little from each other. 
 
 The invention of the logarithms has been attributed to Chriftianua 
 Longomontanus, one of Tycho Brahe's difciples, and an eminent aflro- 
 noraer and mathematician in Denmark. The hackneyed flory which 
 gave rife to this, is told by Anthony Wood in his Athence Oxon'tenfes *, 
 and is as follows ; " Gne Dodlbr Craig, a Scotchman, coming out of 
 ** Denmark into his own country, ca led upon John Neper baron of Mer- 
 " chifton near Edinburgh, and told him among other things of a nev^ 
 " invention in Denmark (by Longomontanus as 'ti>s faid) to fave the 
 ** tedious multiplication and divifion in allronomical calculations. Ne- 
 ** per being follicitous to know farther of liim concerning this matter, 
 *' he could give no other account of it than that it was by proportional 
 " numbers. Which hint Neper taking, he defired him at his return to 
 • call again upon him, Craig alter fome weeks had palTed did fo, and 
 
 " Neper 
 
 • VoL I. p. 469.
 
 INVENTIONS OF NAPIER. ^i 
 
 *• Neper then fliewcd him a nide draught of what he called Canon Mi' 
 ** raii/is Logarithmorinn : which draught with fome alterations he prin- 
 " ting in 1 6 14, it came forthv^ith into the hands of our author Briggs 
 " and into thofe of Will. Oughtred, from whom the relation of this 
 " matter came." 
 
 This (lory is either entirely a ficflion, or much mifreprefented. There 
 is no mention of it in Oughtred's writings *. There are no traces of 
 the logarithms in the works of Longomontanns f , who was a vain maa 
 and fui-vived Napier twenty nine years J, without ever claiming any- 
 right to the invention of thofc numbers, which had for many years 
 been univerfally ufed over Europe,- 
 
 The following hypothefis may perhaps obviate the ftory of Anthc* 
 ny Wood. Might not Craig, whom reafon and Tycho Brahe could 
 not dived of the prejudices of the Arillotelian philofophy whlcli.he had 
 imbibed, on returning to Edinburgh from Denmark, vifit Napier and 
 tell him among other literary news that Longomontanus had invented 
 a method of avoiding ti.e tedious operations of multiplication and divi- 
 fion iu the foiution of triangles ? After getting the beft anfwers this 
 dodlor could give to Napier's queries relative to this method, I per- 
 ceive, fays the baron of Mercliirton, that Longomontanus hath inven- 
 ted, improved, or Itolcn from the Furidamentum j/ijlronom'tcum^ the Prof- 
 thaphscrefis of Raymar : but if you will. take the trouble of calling upon 
 
 me 
 
 • Oughtred's Clavis Math, Oxon i(^77i &c. \ Smith, Briggii vita, and "Ward's h'vcs. Axi, 
 Briggs. 
 
 X VoITuis (dc Nat. Aitiuci) cited Ky Vv'aid, places the death of Longomontanus in the year 1647.
 
 (^i L I r E, W R I T I N G S, A N D 
 
 ine fomc time hence, I will fliew you a method of folving triangles by 
 proportional numbers quite difUndl from this we have been talking of; 
 "which method came into my head fome fhort time ago, and will re- 
 quire many years intenfe thinking and labour to bring it to perfedion. 
 Accordingly a few weeks afterwards, when Craig returned to Merchi- 
 fton, Napier fhewed him the firft rude draught of the Canon Mir'tficus, 
 Craig, having occafioii to write very foon after to Tycho Brahe, men- 
 tioned to him tliis work without faying any thing about its author *. 
 
 Justus Byrgius alfo, inftrument maker and aftronomer to the Land- 
 grave of HefTe, a man of real and extraordinary merit, is faid by Kep- 
 ler, in his I'abul^z Rudolpha^ to have made a difcovery of the Loga- 
 rithms, previous to the publication of the Canon Mir'ificus. The paf- 
 fage referred to is as follows : " Apices logiflici, Jufto Byrgio, multis 
 annis ante editionem Nepeiranam, viam praeiverunt ad hos ipfifTimos 
 logaritlimos (i. e. Briggianos) etfi homo cunclator et fecretorum fuorum 
 cuflos, fxtum in partu deftituit, non ad ufos publicos educavit. That is 
 the accents (', ", "', "", Sec. denoting minutes, feconds, thirds, fourths, 
 &.C. of a circular arch) led Byrgius to the very fame logarithms (now in ufe) 
 ^ many years before Napier s tjoork appeared: but Jiflus being indolent and re-- 
 ferved (or jealous) "with regard to his own inventions ^forfook this his offspring 
 (at or) in its birth ^ and trained it not up for public fervice. 
 
 It 
 
 * Nihil autem (writes Kepler to Cnigerus) fupra Ndpeiranam rationera efTe puto : etfi quidera, 
 
 Scotus quidam, literis ad T)-chonem anno 1594, fcriptis, jam fpcm fecit Canonis lUius Mirifici. KepL 
 
 Epift. a Gotthcb. Hantfch. foUo p. 460. 
 
 f Thus Bj-rgiui mijjhi conceive Log. a" = o 
 
 Log. a' = I 
 Log. a" ■=. 2 
 tog. a"'= 3 &c « being any number kfs than €<a.
 
 INVENTIONS OF NAPIER. 53 
 
 It may be obfcrved that this afFair rcfls on the fingle teflimony of 
 Kepler ; but it would perhaps be coiifidered as a fpecies of herefy to 
 doubt the teflimony of fo great a man. It has been infinuated, how- 
 ever, that from partiality to a countryman lie might imagine he faW 
 more than was really to be found in the papers of Byrgius *. Indeed 
 the cxpreflion, frtum in part u dejiituit^ gives a colour of truth to the in- 
 linuation, and tempts one to think, that Julius' acquaintance witli the 
 logarithms, was much on a par with that of Stifellius. Moreover, it 
 is highly probable, that even the profound and penetrating Kepler might 
 have perufed the manufcripts of Byrgius, without paying any particu- 
 lar attention to his principles of the logarithms, had he himfelf not been, 
 previoufly acquainted with Napier's theory of thofe numbers. Neitlier 
 does it feem probable that Byrgius, had he known its value, could have 
 been fo indolent, fo unreafonably referved, and fo dead to the fenfe of 
 reputation, as to conceal from all the world an invention fo ufeful and 
 fo glorious. AVe know alfo, that he communicated to his fcholars and 
 others a mod ingenious and eafy method of conflrucfling the tables of the 
 natural fines f . But fetting all this entirely afide, and granting a great 
 deal more in favour of Byrgius than Kepler's words impute to him ; 
 nothing can thereby be dctracfled from the merit of Napier, who never 
 faw or heard of Byrgius' pretended difcovery of the logarithms ; for, 
 by Kepler's own confeffion, homo cuntlator et fecretorum fuorum cujlos^ hoc 
 invcntum non ad ufiis publicos educavit. 
 
 O It 
 
 • Nfontucla Hifloire des MathcmatiqiiCJ. 
 
 \ This method is unfolJi;d, and dedicated to Juftiis Bvrcfiiis its int^ntor by Rnywar in f.is Fu'Js- 
 
 nciiiu::, '"■ ■ •-•>//. Sec alfo a part of a letter of R'Hhmanuus to Tytho Br.i'..'-- in Ci'l'.'-'' V^f. 
 Tycli. 9'
 
 j4 L I F E, W R I T I N G S, &c. 
 
 It is therefore upon clear and indubitable evidence that, cum de ali'u 
 fere omnibus praclarh inventh pliires ' contendant gentes^ omnes Neperum loga- 
 rUhmorum antborem agnofcunt qui tanti inventi gloria folus fine eemido fruitiir * ; 
 while feveral nations contend for almofl every other famous invention, 
 all agree in recognifing Napier as the unrivalled author of the loga- 
 rithms, and as folely entitled to the glory of fo great a difcovery. 
 
 SECTION 
 
 Keil de Log. Prstf.
 
 SECTION IV. 
 
 NAPIEr's method of constructing the LOCAHITHMIC CAHOH-i 
 
 Had Napier's principal idea been to extend liis logarithms to all arith- 
 metical operations whatever, he would have adapted them to the fcrics 
 of natural numbers, i, 2, 3, 4, &c. In that cafe, having coniidcrcd 
 the velocity of the two points as continmng the fame for a very fmall 
 fpace of time, after fetting out from N and L (Fig. XL), he would have 
 taken Nn itfelf as tlie logarithm of CN + Nn, or Cn. Now as Cn fur- 
 pafTes CN or unity by a very fmall quantity, it is evident that, when 
 raifed to its fucceflive powers, there will be found in the feveral pro- 
 ducts fuch as are very near in value to the natural numbers i, 2, 3, 4, 
 &c. agreeably therefore to the above theory (Sedl. III. prop. 9.) Nn be- 
 ing eq\ial to d, and x being a politive integer, any natural number may 
 be reprefented by (i+d)" and its logarithm in Napier's fyftem by xd. 
 
 By this formula might the logarithms of all the primary numbers 3, 
 5, 7, &c. be calculated ; from wliich thofe of all the compofite numbers 
 4, 6, 8, 9, ID, &C. are ealily deduced by fimple additions (Se(5t. IIL 
 prop. 7.) or by multiplications by 2, 3, 4, 5, Sec. (Sed. III. prop. 9), 
 
 Napier's
 
 S6 LIFE, WRITINGS, and 
 
 Napier's views were entirely confined to the facilitating of trigono- 
 metrical calculations. This is the realbn of his calculating only the lo- 
 garithms of the fines ; the log. of any given number being eafily de- 
 duced from thefe by means of a proportion. 
 
 In order to effed his purpofe, he confidercd that the radius, or fine 
 total, being fuppofed to confift of an infinite number of infinitely fmall 
 and equal parts,' all the; other fines would be found in the terms of a 
 geometrical feries defcending from it to infinity ; and that the logarithm 
 of the radius being fuppofed equal to zero, the logarithms of all the fe- 
 ries, beginning with the radius, would be found in the terms of an a- 
 rithmetlcal feries, afcending from zero to infinity by fleps equal to 
 the logarithm of the ratio in which the geometrical feries defcends. 
 
 Agreeably to this idea, he fuppofes the radiusrr CN=: looooooo, 
 and firll conftrucls three tables, of which the firft contains a geometrical 
 feries defcending from the radius to the hundredth term in the ratio of 
 looooooo to 9999999. It is formed by a continual fubtradling, from 
 the radius and every remainder, its loocooooth part. The decimals 
 in every term are pufhed to the feventh place : a fpecimen of this table 
 follows. 
 
 10000000
 
 INVENTIONS OF NAPIER. 
 
 ^S 
 
 lOOOOOOO . ooooooo 
 I . o.oooooo 
 
 9999999 . ooooooo 
 
 9999999 
 
 9999998 . 000000 1 
 999999B 
 
 9999997 . 0000003 
 9999997 
 
 9999996 . 0000006 
 
 raid fo on to 
 9999900 . 0004950 
 
 The fecond table contains a geometrical fcries dcfcending from the 
 radius to the fiftieth term, in the ratio of 1 00000 to 99999, nearly 
 tqual to that of the firfl term i ooooooo . ooooooo to the laft 9999900 
 . 0004950 of the firft table. It is formed by a continual fubtra(5ling, 
 from the I'adius and every remainder, its 1 00000th part. The deci- 
 mals are pufhed to the fixtli place. A fpecimen of tlus table follows. 
 
 1 0000000 . ooooooo 
 100 . cooooo 
 
 9f}r)999oo . 000000 
 
 99 • 999000 
 
 9999800 . ooiooo 
 
 99 . 998000 
 
 9999700 . 003000 
 
 99 • 997000 
 
 9999600 . 006000 
 
 and i'o on to 
 
 9995001 . 222927 
 
 P 
 
 The
 
 «;« 
 
 LIFE, WRITINGS, ano 
 RADICAL TABLE. 
 
 FIRST COLUMN. SECOND COLUMN. 
 
 NATUSAL. 
 
 lOOOOOOO . ocoo 
 9995000 . 0000 
 9990002 . 5000 
 
 9985007 . 4987 
 
 9980014 . 9950 
 
 and fo on to 
 9900473 . 5780 
 
 ARTIFICIAL. 
 
 
 
 
 5001 . 
 10002 . 
 
 2 
 
 5 
 
 15003 . 7 
 20005 • 
 
 and fo on to 
 
 100025. 
 
 II 
 
 NATURAL. 
 
 9900000 
 9895050 
 9890102 
 9835157 
 9880219 
 
 and fo on to 
 9801468 . 8423 
 
 0000 
 0000 
 4750 
 4237 
 8451 
 
 and fo on to 
 COLUMN 69. 
 
 NATURAL. 
 
 5048858 
 
 5P4'53.^3 
 504381 1 
 
 5041289 
 
 3038768 
 
 and fo on to 
 
 499^6^9 . 4034 
 
 8900 
 46.5 
 2932 
 3879 
 7435 
 
 ARTIFICIAL. 
 
 6834225 
 
 68^ 
 
 9227 
 
 8 
 
 I 
 
 3 
 6 
 
 S 
 
 and fo on to 
 6934250 . 8 
 
 684422S 
 6849229 
 685423; 
 
 ARTIFICIAL. 
 
 100503 
 105504 
 110505 
 
 I 15507 
 120508 
 
 and fo on to 
 200528 . » 
 
 The numbers and logarithms in the above table, coinciding nearlj^ 
 •u'ith the natural and logarithmic fines of all the arcs from 90° to 30°, 
 he was enabled, by means of prop. 16. or 1 7. an.l a table of the natural 
 fines, to calculate the logarithmic fines to every minute of the laft 60° 
 of the quadrant. 
 
 In order to obtain the logarithms of the fines of arcs below 30°, he 
 propofes two methods. 
 
 The firfl is this. He multiplies any given fine of an arc lefs than 
 ■go® by the number 2, 10, finding the logarithms of the numbers 2 
 
 and
 
 INVENTIONS o? NAPIER, ^ 
 
 and lo by means of the radical table, or takes fome one of the com- 
 pounds of thefe fo as to bring tae product within the compafs of thd 
 radical tabic. Then having found, in the manner before defcribed, 
 the logaritlun of this product, he adds to it the logarithm of the mul- 
 tiple he had made ufe of; the ium is the logarithm ibught. 
 
 The fecond method is derived from a property of the fines which 
 he demonflrates. The property is tliis : Half the radius is to the fin6 
 of half an arc, as the fine of the complement of half that arc is to the 
 fine of the whole arc. Hence, as is evident from a foregoing prop, 
 that the logarithm of the fine of half an arc may be had by fubtradting 
 the logarithm of the fine of the complement of half that arc from the 
 fum of the logarithms of lialf the radius and of the fine of the whole 
 arc. 
 
 By this fecond method, which is much eafier than the firft, the lo- 
 garithms of the fines of the arcs below 45° may be obtained ; thofe 
 above 45° having been calculated by help of the radical table. 
 
 The logarithms of the fines to every minute of the quidrant being 
 found by the ingenious methods above defcribed, the logarithms of 
 the tangents were eafily deduced by one fimple fubtrailion of the lo- 
 garithm of the fine of the complement from that of the fine for each 
 arc. The logarithm of the radius, which fo frequently occurs in tri- 
 gonometrical folutions, having been very advantageoufly made equal 
 to zero, the logarithms of all the tangents of arcs below 45° and of 
 fill the fines muft have a dillcrent fign from that of the logarithms of 
 
 (^ all
 
 ^ LIFE, WRITINGS, and 
 
 all the tangents of arcs above 45°. Napier chofe the pofitlve fign for 
 the former which he calls ahnudantes^ and the negative for the latter 
 which he calls dcfeLl'tv'u 
 
 The arrangement of the numbers in Napier's logarithmic table, is 
 nearly the fame with that neat one which is flill in ufe. The natural 
 and logarithmic fmes and the logarithmic tangent of an arc and of its 
 complement ftand on the fame line of the page. The degrees are coni- 
 tinucd forwards from 0° to 44° on the top, and backwards from 45? 
 to 90° on the bottom of the pages. Each page contains feven columns ; 
 the minutes defcend from o' (to 30' or from 30') to 60' in the firfl, and 
 from 60' (to 30' or from 30') to o' in the lafl of thcfe colmnns. The 
 natural fines of the arcs, on the left and on the right hand, occupy the 
 fecond and fixth column, and their logarithms the tliird and fifth ra- 
 lpe6tively. The fourth column contains the logarithms of the tangents 
 which are taken pofitively if they refer to.th& arcs on the left, and ne- 
 gatively if they refer to the aaxs on the right hand. A fpecimen of 
 this table, is here, annejted. . 
 
 Cr.
 
 INVENTIONS OP NAPIER. 
 
 6i 
 
 Gr. 
 
 44 
 mi. 
 
 SINUS. 
 
 LOGARITIIMI. 
 
 DIFFERENTIA. 
 
 LOGARITHMl. 
 
 SIMUS. 
 
 
 3^ 
 31 
 32 
 
 7--''>-y3 
 7CI1I67 
 
 701^24' 
 
 35J57'J7 
 
 3550808 
 
 3547^P 
 
 «745-i' 
 168723 
 162905 
 
 3379226 
 3382085 
 3^84946 
 
 7'3i504 
 7130465 
 7128225 
 
 30 
 
 29 
 
 28 
 
 i3 
 34 
 
 3) 
 
 7^'5<'4 
 70173S7 
 7019459 
 
 3344^95 
 
 354'^9+> 
 35??9S9 
 
 157087 
 151269 
 •-I545' 
 
 338780S 
 3390672 
 
 3393537 
 
 7126385 
 
 7' 24:44 
 7122303 
 
 27 
 26 
 
 25 
 
 36 
 
 37 
 38 
 
 70:1530 
 7023601 
 7025671 
 
 3?33o^9 
 353= '42 
 
 »3'J'^33 
 ihS'4 
 i2;9-y<S 
 
 3396406 
 
 3399275 
 3402146 
 
 7120261 
 7118218 
 71 16)75 
 
 24 
 
 23 
 
 22 
 
 39 
 40 
 41 
 
 7027741 
 7029810 
 7031879 
 
 3527'9; 
 35M-4J 
 352'3" 
 
 122178 
 116359 
 I 10541 
 
 3405019 
 
 3407 P94 
 34 "3770 
 
 7"4'3' 
 71 12086 
 71 1004 1 
 
 21 
 20 
 
 '9 
 
 42 
 43 
 44 
 
 -'=33947 
 7036014 
 7..3!^o8i 
 
 35'«37' 
 35"5432 
 35'2-'95 
 
 •04723 
 
 9^'V"4 
 9;S,-6 
 
 34'364« 
 34'552« 
 3419409 
 
 7107995 
 7105949 
 7103901 
 
 18 
 
 17 
 16 
 
 45 
 46 
 
 47 
 
 7040147 
 7042213 
 7044278 
 
 3509560 
 3506626 
 3 503'^9* 
 
 8i4;o 
 
 75^32 
 
 3422292 
 3425176 
 342 8o<.2 
 
 7101854 
 7099'so6 
 7097757 
 
 '5 
 '4 
 
 '3 
 
 48 
 49 
 50 
 
 704^342 
 7048406 
 7050469 
 
 35C0764 
 
 3497S35 
 34s>4yo-'' 
 
 6y824 
 64OC6 
 5^,78 
 
 3430940 
 
 343?W29 
 3436730 
 
 7095708 
 7093658 
 7091607 
 
 12 
 1 1 
 
 10 
 
 5' 
 
 53 
 
 7052532 
 
 7354J94 
 70566; s 
 
 34919^3 
 348901.0 
 34861:9 
 
 52360 
 
 4^'543 
 
 40725 
 
 3439623 
 34425'7 
 34454' 3 
 
 70^9556 
 7087504 
 
 7CS5452 
 
 9 
 8 
 
 7 
 
 54 
 55 
 56 
 
 7058716 , 
 7060776 
 7062 3 S6 
 
 34«3^'9 
 3480301 
 
 34-7^^5 
 
 34'J-« 
 
 2.^0;o 
 2327? 
 
 344^3" 
 345'2" 
 3454"2 
 
 7i«3i99 
 70S1345 
 70-9291 
 
 6 
 
 5 
 4 
 
 57 
 58 
 59 
 
 70046^5 
 7066953 
 
 7069011 
 
 34V4-173 
 
 34755S7 
 3.68645 
 
 '7455 
 116,7 
 
 58. 8 
 
 3i570'5 
 34599-0 
 
 ^4'.2K>7 
 
 7077236 
 
 70;5i8i 
 7071 1 25 
 
 3 
 
 2 
 I 
 
 Co 
 
 707 1000 
 
 34'^j7if 
 
 
 
 3465735 
 
 7071008 
 
 
 
 45 
 mi. 
 Gr. 
 
 In the Appendix to the Canonls mirif.cl coiijlnt^lo, Napier delivers three 
 other methods of computing the logarithms ; but as thcfe methods are 
 generally better adapted to the conflructloa of a fpecies of logarithms 
 dillcrent from that I have defcribed, I fhall poftpone the account of 
 them to the next fcdioa. 
 
 Tin
 
 '^i L I F E, W R ! T I N G S, «ce. 
 
 The ingenious method by which Napier conftruftcd the radical ta- 
 ble is almoft peculiar to the fpecies of logarithms it contains : It does 
 not feem, however, to be fufceptiblc of all the accuracy one would wilh ; 
 for, notwithflanding the many precautions he had taken, particularly 
 in pufliing his numbers to feveral decimal places, the logarithms in his 
 canon often differ from the trutli by feveral units in the lail figure. Of 
 this he himfelf was apprifed by finding different refults from the two 
 methods of determining the logarithmic fmes of arcs xmder 30°. In or- 
 der to remedy this defcdl, he propofes adding another zero to the radi- 
 us ; by which means, in purfuing this fame method, the logarithms of 
 the fines might be obtained true to an unit in the eight figure. 
 
 SECTION
 
 SECTION V. / 
 
 THE COMMON LOGARITHMS DEVISED BY NAPIER AND PREPARED B^ 
 BRIGGS, AND THE METHODS PROPOSED BY NAPIER FOR COMPU- 
 TING THEM. 
 
 One capital difadvantage attending the fpecies of logaritlirtis which 
 firft occurred to Napier, arifes from the difference between the fign of 
 the logarithms of the tangents of arcs greater than 45° and the fign of 
 the logarithms of the fines of all the arcs of the quadrant. 
 
 This defecl was eafily remedied by fuppofing the fmallefl pofublc 
 fine equal :o i and its logarithm o ; as in this cafe, the logarithms of 
 all tlie fines and tangents of every arc in the quadrant would have the 
 fame fign. But, if the fame fpecies of logarithms is made ufe of, the 
 logarithm of the radius, which occurs fo frequently in trigonometrical 
 folutions, would be a number difficult to be remembered. More, there- 
 fore, would be lofh tlian gained by this alteration. What fpecies of lo- 
 garitlxms will free us from a difference in the figns, and at the fame time 
 afford a logarithm of the radius that fliall be eafiiy remembered and 
 eafily managed ? It was tliis very queflion, in all probability, diat led 
 to the common logarithms, which, of all others, are the beft adapted 
 
 R to
 
 64 L r r r, W R I T I N G S, A K D 
 
 to our modern arithmetical notation. This fyflcm of logarithms has 
 for its bafis i as logarithm of tlie ratio of lo to i : i'o that the powers r,. 
 lo, I oo, I GOO, 8cc. of the nvimber i o have their refpe6live logarithms 
 o, 1, 2, 3, Sec, * Here, by the bye, it miay be obferved, that not only- 
 Napier's manner of conceiving the generation of the logarithms, but 
 his having computed that fpecies of logarithms, wluch has been dif- 
 cribed, before the common logarithms occurred to liim, // a convincing 
 proof of his not taking the. bint of the logarithms from the remark of Stijellius^ 
 formerly mentioned. I think it is even beyond doubt that Napier,, in 
 common with all other arithmeticians acquainted with the Arabic, or 
 rather Indian figures, had obferved that the produft of any power of 
 tlie number lo by any other power of that number, was formed by- 
 joining or adding the zeros in the one to thofe in the other ; and tiiat 
 the quotient of any one power of that number by any other, was for- 
 med by taking aviray or defacing a number of zeros in the dividend e- 
 qual to the number of zeros in the divifion ; and all this without think- 
 ing that he was, at that time, making the fundamental remark of the 
 logarithms. Nor will this feem at all furpriling to thofe who are ac- 
 quainted with the hiftory of fcience and of the human mind. It is fel- 
 dom that -we direcTcly arrive at truth by the moll natural and cafy path; 
 
 Perhaps 
 
 * "W-fi haTtfcen Se£l. III. that in Napier's (j-ftem the velocity of the two moveable points in N and L 
 Fig. XI. is equal and that the logarithm (LI)" of .any number (CN+Nn)^ or 1.0000000,1 )■< is neart- 
 ly equal to (Nn)" or [.ocoooco,i]* In the common fyftem the velocity at L is lefs than the half 
 of the velocity at N; and the logarithm LI of the number [CN-|-Nn]>; [or 1.0000000,13^ is ncaily 
 equal [0.4342945] NnXx or [0.0000000,0434,2945]"= for in making this fuppofition the logarithm 
 of 10, is found to be 1. The logarithms therefore in Napier's fyftem are to the corrcfpondent one* 
 in the common fyftem as i is to o. 4342945 or, what is the fame thing the common logarithms ar« 
 to tbofe of Napier as 1 is to 2.3025S51.
 
 INVENTIONS or NAIPER. 65 
 
 Perhaps the ftrongefl mark of the greatncfs of Napier's genius is not 
 his inventing the logarithms, but his manner of invendng them. But 
 to return; In this new fyftem the radius was made equal to the roth 
 power looooooQOOo of the number 10, of wliich the logarithm in tlic 
 new fcale is 10. The divifion of the radius into fo great a number of 
 parts, render the fine of the fmalieft feniible arc greater than 1, of 
 which the logarithm is zero : confequcntly, the logarithms of all tlvc 
 fines and tangents of the arc3 of tlie quadrant, being on the fame fide 
 of zero, have the fame fign. 
 
 With regard to the logarithm of the radius, its being cafily mana- 
 ged is fujQlciently obvious. 
 
 Thus in our common logarithms the difadvantagcs of Napier's fyftem 
 are avoided, whilft its advantages ai'e retained and united to feveral 
 otiiers. Of thefe additional advantages in the common canon, tlic 
 mod capital is, that the units in the firft figure (to which Briggs gave 
 the name of charadleriflic) of the logarithm are fewer by one than the 
 figures of the number to which that logarithni corrcfponds. 
 
 Whether Napier, or Briggs,^/)'? imagined this newfpecies of log** 
 jithnis, is a queftion which the learned do not feem as yet to have per- 
 ft<^ly decided. 
 
 The only evidence we have on which a decilion can be grounded, 1» 
 Contained in the following paiticdarsr 
 
 I,
 
 6^ L I F E, W R I T I N G S, A N D 
 
 I. In a letter to Uflier afterwards Archbifhop of Armagh dated the 
 loth of March 1615, the year after the pubhcation of Napier's Canon. 
 Biiggs -writes thus *, " Napier lord of Merchiflon hath fet my head 
 *' and hand at work with his new and adiTiirable logarithms : I hope 
 •* to iee him this fummer if it pleafe God ; for I never faw a book 
 •* which pleafed me better, and made mc more wonder." 
 
 II. In the dedication of his Rabdologia, publifhed 161 7, Napier has 
 the following words, " Atquc hoc mihi fini propofito, logarithmorum 
 *' canonem a me longo tempore elaboratum fuperioribus annis edendum 
 *' curavi, qui reje<51is naturalibus numeris, ct operationibus quce per 
 " eos Hunt, difiicilioribus, alios fubllituit idem praeftantes per faciles 
 " addtiones, fubflracliones, bipartitiones, et tripartitiones. Quorum 
 " quidem logarithmoi-um y^m^/w aliain multo pr^ftantiorem nunc etiam in- 
 *!* venimus, et creandi methodum, una ciun eorum ufu (fi Deus lon- 
 " giorem vitx et valetudinis ufuram conceflcrit) evulgare ftatuimus : 
 Vipfam autem novi canonis fupputationem, ob infirmam corporis noftri 
 V jraletudinem, viris in hoc iludii genere verfatis relinquimus : impri- 
 " mis vero docliillmo viro D. Henrico Briggio Londini publico Gco- 
 *' metric ProfclTori, et amico mihi longe charifllmo". 
 
 • . Ill, In the preface to the logaritbmoruvi cbilias primay a table of the 
 
 common logarithms of the firft thoufand natjiral members, Briggs ex* 
 
 prefles himfelf in the following terms ; " Why thefe logarithms differ 
 
 ** from thofe fet forth by their illuflrious inventor, of ever refpedful 
 
 *' memory, in his canon mirlficus^ it is to be hoped, his polthumous wor^ 
 
 f* will fhortly make appear." 
 
 IV. 
 
 • The life of Archbifnop Ulher and his conefpondencc, by Richard Par, D. D. 1686. folio, 
 f age. 36.
 
 INVENTIONS OF NAPIER. 67 
 
 IV. In the preface the Arithmetica Logarithmeca *, there is the fol- 
 lowing paragraph, " Quod hi logarithm! diverii funt (writes Briggs) al> 
 " iis quos clarilfimus vir bare Merchiftonii in fuo edidit canonc mirifi- 
 " CO, non ell quod mireri, enim meis auditoribus Londini publice in 
 " Collcgio Greftiamcnfi horum dodlrinam explicarem; animadverti mul- 
 " to futurum commodius, fi logarithmus finus totius fervxtur o zero 
 " (ut in canone mirifico) logarithmis autem partii decima: ejufdem finus 
 *' totius, nempe finus 5 grad. 44 min. 21. fecund. elTet 1.00000,00000: 
 " atque ea de re fcripfi flatim ad ipfum, Authorem, et quamprlmum 
 ** hie anni tempus, et vacatloneni a publico docendi munere licuit, 
 " profedlus fum Edinburgum ; ubi humaniilime ab eo acceptus hxG. 
 *' per integrum mcnfem. Cum autem inter nos de horum mutationc 
 ** fermo haberetur ; Ilk fc idem dudum finfijfe^ et capuifTe dicebat : vc- 
 " runtamen iflos, quos jam paraverat, edendos curafTc, donee alios, fi 
 '* per negotia et valetudinem liceret, niagis commodos confccifTtt. If- 
 " tarn autem mutationem ita faciendam cenfebat, ut o eiTet logarithmus 
 " unitatis et i ,00000 . 00000 finus totius : quod ego longe commodifli- 
 " mum efTe non potui non agnofcere". " Capi igitur ejus hortatu, rc- 
 " je<5lis illis quos antea paraveram, de horum calculo ferio cogitarc, et 
 '* fequenti aeftate iterum profe(Slus Edinburgum, horum quos hie cxhi- 
 " beo praecipuos, illi oflendi. Idem etiam tertia jefbite libentiflime fac- 
 " turns, fi Deus ilium noBis tamdiu fuperftitem cfTe voluifTet f." 
 
 It may here be obfcrved, that the manner in which Briggs propofed 
 the application of the common logarithms to trigonometrical purpofes, 
 
 S did 
 
 • Publifhed in 1624. 
 
 •f Ulacg in his title page to his edition of Bngg's log. Trrites to the fame purport. " IIos cumerM 
 " primiv inucnit clarifllmus vir Jo<innes Nfperus Baro Mercliillonii j toi auttin ex cjufJon/inUntiii, mw- 
 *• tavilt eorutnquc ortum ct ufum illullralt Ilcnricus Briggius".
 
 68 L I F E, W R 1 T I N G S, A N D 
 
 (lid not at all tend to obviate the chief difadvantace of Napier's Canon : 
 For according to Briggs' idea tlie fign of the logarithms of the fines and 
 the tangents lefs than the radius muft be the oppofite of the fign of the 
 logarithms of the tangents greater than the radius. It fcems probable,, 
 therefore, that Briggs had been led to the common logarithms in en- 
 deavouring to get rid of the indirccfl method of finding the logarithiiig 
 of the natural numbers by means of Napier's logarithmic Canon. 
 
 From the extracSls above given it appears that the common Icgaritlims 
 had occurred to Napier before they occurred to Briggs : For the mo- 
 defly and integrity of Napier's character put beyond difpute tlie truth 
 of what he mentioned to Brigg's at tlieir firfl meeting, and to the Earl 
 of Dunfermline in the dedication of the Rabdologia. But if the ha- 
 ving firft communicated an invention to the world be fufficient to enti- 
 tle a man to the honour of having firft invented it, Briggs has a better 
 title than Napier to be called the inventor of this happy improvement of 
 the logaritluns *. For Briggs mentioned it to his pupils in Grefhanv 
 College before the publication, in 1616, of Edward Wright's tranlla- 
 tion of [the Cancfj M'lrificus^ in the Preface to which Napier gave the 
 firft notice of xK\% improvement. With regard to the paflage in the 
 preface to the Ch'illas prima publiflied after Napier's death, where Briggs 
 feems to require an acknowledgment from th^editor of the Canonis rr.i- 
 rifici conftru(5lio, that he had alfo imagined the new logarithms ; the 
 overfight or fault hes at the door of Napier's fon and not at his own. . 
 Had Napier lived to publifh his laft mentioned work, it is hardl)- poC- 
 fible to entertain a fiiadow of doubt, but that he would have done am- 
 
 • Hutton Math. Tab.
 
 INVENTIONS OF NAPIER. 69 
 
 pie juftice to ISriggs in this particular. Napier and Briggs had a reci- 
 procal efteem and afFediou for each other, and there is not the fmaileft 
 evidence of there having cxiiled, in the brtaft of cither, the leaft parti- 
 cle of jealoufy j a pafiion unbecoming and diigraceful in a man of 
 merit. 
 
 We fhall difmifd this affair -with obfcrving, i. That after the inven- 
 tion of the logarithms, the difcovery of the bcft fpecies of logarithms wat 
 no difBcult affair : 2. That the difcovery of the common logarithms at" 
 that time, was a fortunate circumflance for the world, as there are few 
 pofleHed of ing-enuity and patience fuificient for the confcnKTuon of 
 fiich exteniive and accurate tables as are thofe of Briggs' Arithtnctica 
 hgar'ithmica ; and 3. That the invention of the new fpecies of loga-* 
 rithms is far from being equal to feme other of liriggs' inventions. 
 
 We come now to give a very brief dcfcripfion of thofe other methods 
 of conftru6llng the logarithms, propofed by N.ipicr in the appendix to 
 liis poflliumous work. 
 
 The firfl of thefe methods is the following: The logarithm of i be- 
 ing fuppofed o, and the logarithm of 10 i followed by any number 
 of zero, 1 0000000000 fof* example ; this lalt logai'ithm and the fuccef- 
 five quotients divided (ten times) by the numtjer 5 will give thefc (ten) 
 logarithms 2000000000, 400000000, 80000000, 16000000, 3200000, 
 640000, 128000, 25600,5/20, 1024 to which the re fpedlive corrcfpon- 
 dcnt numbers may be found by extra<5ling the 5th root, the 5th root 
 of the 5th root, the 5th root of the 5th root of the 5th root, Sec. of 
 
 the
 
 7<y 
 
 LIFE, WRITINGS, and 
 
 the number lo. Then the laft logarithm 1024 and the fuccefllve quo- 
 tients divided (ten times) by the number 2, will give thefe (ten) loga- 
 rithms 512, 256, 128, 64, 32, 16, 8, 4, 2, I, &c. to which the refpec- 
 tivc correfpondent numbers may be found, by extrading the fquare 
 root, the fquare root of the fquare root, the fquare root of the fquare 
 root of the fquare root, &c. of the number (foimd as above dire(5led) 
 correfponding to the logarithm 1024. By addition thefe (twenty) lo- 
 garithms, and by multiplication their refpedlive natural numbers fervc 
 for finding a great many other logarithms and their numbers. 
 
 The fecond method is this: The logarithms (oand loooooooooo for 
 example,) of any two numbers i and 10 being given, the logarithm 
 of any intermediate number (2 for example) may be found by taking 
 continually geometrical means, firft between one of thefe numbers (10) 
 and this mean, then between the fame number (10) and the laft mean, 
 and fo on till there be found the number (2) wanted ; of which the lo- 
 garithm will be the correfponding arithmetical mean (3010299957) be- 
 tween the two given logarithms (o and 1 0000000000). 
 
 The third method is as follows : Suppofe the common logarithm of 
 a number not an integral jxjwer of 10 (2 for inftance) find the num- 
 ber of figures in the loth, 1 00th, loooth, &CC. power of that number ; 
 The fuccefllve numbers of figures (4, 31, 302, 301 1, 8cc.) in thefdpow- 
 cfs (2'°, 2"", 2' °°, 2'°°°°, &c.) will always exceed by lefs than unity, 
 but continually approach to the logarithm [30102099566, 8<c.] re- 
 quired. 
 
 Ths
 
 INVENTIONS OF NAPIER. 71 
 
 Th2 Crfl of thefe methods is very operofe, andby itfelf infufficlent 
 for conflru(5ling a complete logarithmic canon. The other two are 
 much preferable. The laft is particularly well adapted for finding the 
 logarithms of the lower prime numbers : For, fmce the number of fi- 
 gures in the produd of two numbers, is equal to the fum of the num- 
 ber of figures in each fador ; except the pi'oducl of the firfl figures in 
 each faclilor be exprefTed by one figure only, which often happens ; a 
 few of tlie firft, or left hand figures of the confecutive tenth powers 
 of the given number, will fuffice for finding the number of figures ixi 
 thefe powers. 
 
 This laft method depends on the diftinguifhing property of the com- 
 mon logarithms, v/hich is, as was formerly obferved, that the units in 
 [x] the rational logarithm of a number [10'*] are one fewer than the 
 number of figures in that number [10']. Whence it follows, that 
 the units in the irrational logarithm of any other number are not quite 
 one fewer than the number of integral figures in this other number. 
 Now, as in a feries of continued proportional numbers, the rcfult of 
 any two terms is the fame, if one of the terms is raifod to the power 
 indicated by the exponent of the other, or if this other is raifed to 
 the power indicated by the exponent of the firfl ; any number raifed 
 to the power indicated by the logarithm of 10 is equal to 10 raifed to 
 the fewer indicated by the logarithm of that number. If, therefore, 
 [the logarithm of 10 being 10000, &c.] Y is any number not an inte- 
 gral power of 10 and y its logarithm, wc fhall have Y'°°^°' ''^==ioy» 
 and the number of figures in Y^°°°^' ^'^- will exceed y by lefs than i. 
 
 T SECTION,
 
 SECTION vr. 
 
 THE IMPROVEMENTS MADE ON THE LOGARITHMS. 
 
 1 HE improvements that have been made upon the logarithms after 
 the death of their inventor, regard the theory, the methods of conftruc- 
 tion, the accuracy, extenfivenefs, and form of the tables of thefc 
 numbers. 
 
 However ingenious and beautiful Napier's manner of delivering the 
 theory of the logarithms is, it muft be acknowledged that it labours 
 under one capital impropriety — treating geometrically a fubjecfl whic^ 
 properly belongs only to arithmetic. Senlible of this, Kepler *, Nicolas 
 Mercator t, Halley |, Cotes ||, and other mathematicians of the firft 
 note, have treated the theory of the logarithms in a diiFerent and tx'uly 
 fcientific manner. Their ideas are founded on the definition of the lo- 
 garithms — Numcrl rat'ionem cxponcntcs ; which, although it is not exprefs- 
 ly Napier's, is eafily deducible from his theory. Thus, in a geometrical 
 progrelhon, having any finite number c greater tlian unit^' for it's bafis, 
 the exponent x is tlie logarithm of the ratio of the number c'' to c° 
 
 or 
 
 • Chilias Logarithmorum 1624. Tal>. Rudolph. 1627. f LogarithnK) tecUnia, 1668, 
 
 J Phil. Trana. 1695. || Hnrraonia Mcnfurar. 1722.
 
 74 LIFE, W R I T I N G S, and 
 
 or uuliy : And, if the quotient of two quantities is taken as the meafure 
 of their ratio, the definition is rendered more fimple, and x will be the 
 logarithm of c"- Upon this principle is founded the analytical theory 
 of the logarithms in the appendix:. 
 
 It was chiefly by the two lafl rficthods, defcribed in the foregoing 
 fe(flion, that Briggs conflrucflcd his logarithms. He invented alfo an 
 original method of conflrucfling logarithms by means of the firft, fecond, 
 third, SvC. diiTerences of given logarithms. How he came by it is not 
 known. He dcfcribes it in his arithmetica logarithmica and there is a 
 demonflration of it in Cotcs*s Harmonia, in Bertrand's Mathematiqucs, 
 and in the works of a great many other authors. 
 
 EcMUKD Gxmter, Profcffor of Aflronomy in Grefliam College, who 
 "was the fir/l that publifhed a table of the logarithmic fines and tangenti 
 of that kind which Napier and Eriggs had lafl agreed on, applied, in 
 the year 1623, or 1624, the Icgarithras to a ruler which bears his name. 
 This fcale is of very great ufe in Navigation, and in all the pradlical 
 parts of geometry where much accuracy is not required. On the ac- 
 count of this logarithmical invention, Gunter, after Napier and Eriggs, 
 has the befl claim to the public gratitude. 
 
 ArTEX Napier's death almoft fifty years elapfed before the invention! 
 of the exprefHons of the logarithms by infinite feriefes. Of thefe the 
 three following, from which a great number of others are eafily deri- 
 \-ed, were the firfl. * 
 
 liOgarithiA 
 
 • Appendiis; an. A. lojf,
 
 INVENTIONS OF NAPIER. 7; 
 
 Logarithm of {i-\-a)=a — ^^*+t^'+^c. ----- X 
 Logarithm of (i — a)= — a — ^a^ — ^a' — 8cc. Y 
 
 Logarithm of (^^^^) =a-^W +1^' +Scc, - Z 
 
 These formulae X, Y, and Z will converge the more quickly in pro^- 
 portion as a is fuppoled lefs than unity ; and the fum of a few terms 
 will generally fufEce. They are the values of Napier's logarithms, but 
 will reprefent every fpecies of logarithms by being multiplied by an 
 indeterminate quantity 2/, which is called the modulus of the fyftem. 
 
 The forxnula X was invented by Nicholas Mercator in the year 1667, 
 and publiflied in his Logarlthmotechn'ta the year following. Gregory of 
 St Vincent, about twenty years before, had lliewn that one of the afymp- 
 totes of the hyperbola being divided in geometrical progreflion, its ordi- 
 nates parallel to the other afymptote are drawn from the point of divi- 
 vifion, they will divide into equal portions the fpaces contained be- 
 tween the afymptote and the curve : From this it was afterwar<is point- 
 ed out by Merfennus, that, by taking the continual fume of thofe parts 
 there would be obtained areas in arithmetical progreflion correfpondlng 
 to abfcifles in geometrical progreflion, and confeq\icntly that thefe ar- 
 eas were analogous to a fyllem of logarithms *. Wallis, after this, had 
 remarked that the ordinate correfponding to the abfcis ^, counted on 
 the afymptote of the equilateral hyperbola from a diftance equal to the 
 
 femi-axis i, is equal to — ; — ; and he had demonftrated, in his Aritbme' 
 
 tica hifinitorum publiflied in 1655, that the fum of i"-j-2'"-f-3'"-(- &:C. 
 
 ^-«'" [a reprefenting a finite quantity divided into an infinite num- 
 
 U bff 
 
 • Hottoa's Ma»k Tab-
 
 ^6 LIFE, WRITINGS and 
 
 ^7*^+ I 
 
 bcr of equal parts) is equal to *. With thefe data Mercator fet 
 
 himfelf to find the area correfponding to the abfcifs c, or^ what is the 
 fame thing, the logarithm of ( i + a), which he happily accomplilhed by 
 firfl developing, in the manner now commonly pra(5lifed, the fraction. 
 
 into I — a + a"" — a' + &c. which had not been attempted before : 
 
 then, fuppofing a equal fucceflively to i, 2, 3, 4, 8cc. a, and 
 
 laftly, taking fucceflively the funis of all the zero, firfl, fecond, third,, 
 8cc. powers of thefe nuuibers f . 
 
 In the fame year 1668 James Gregory, in his Exercltationes Gcome- 
 tricce^ gave a demonftration of Mercator's formula for the quadrature 
 of the hyperbola different from his. He demonftrated the formula Y 
 and found the formula Z by fubtra(5ling Y from X. He found too the 
 the value of log. (i — a*) = — a — \a — \a — 8cc. by adding Y to X : but 
 this formula may be looked on as a folecifm when applied to numbers : 
 for the fame refult will be obtained by fuppofing « to be a fquare, in 
 the formula Y, and even a more general refult may be obtained by 
 fuppofing a to be any power of a number.. 
 
 Sir Ifaac Newton, by his general method of the quadrature of curves, 
 greatly fim.plificd that of the quadrature of the afymptotic fpaces of the 
 
 equilateral hyperbola. The ordinate, (being as before = )—= multipli- 
 
 ed by a the fluxion of the abfcifs, becomes the fluxion of the corref- 
 ponding afymptotic area : This produ(ft, developed in the manner of 
 
 • • • • 
 
 Mercator, is a — aa + a a — a'a + Scc. Taking the fluent of each term 
 
 of 
 
 • Montuda Hill, de Math. f Appendbt, Hyperbola..
 
 INVENTIONS OF NAPIER. 77 
 
 of this feries gives the fluent of the area that is the losarichm of 
 I + X equal to X as before. It appears, from a letter of Newton's to 
 Oldenburgh, that Newton had difcovercd the quadrature of the hyper- 
 bola by infinite but perhaps not general feriefes, before the pubhcation 
 of the Logaritbmotechnia'^ . Something of the fame kind had alfo been 
 difcovered by Lord Brouncker f. 
 
 The areas of the equilateral hyperbola, as above defcribed, exhibiting 
 the logarithms of Napier's fyilem, occafioned the appellation hyperbolic 
 to his logarithms. It is dlilicult to accoimt for the propriety of this e- 
 pithct to Napier's logarithms ; lince not only the afymptotlc areas of 
 equilateral but thofe of any other hyperbola may be made to repre- 
 fenr every pofLblc fpecies of logarithms, by fuppoiing, in the fame hy- 
 perbola, the origin of the abfciflcs on the one afymptote at diJi'erent 
 diflances from its interfecftion with the other. Thus the afymptotic ar- 
 eas of the equilateral hyperbola will reprefent the common logarithms, 
 if the origin of the abfciiTes is taken at the point on the afymptote 
 where the ordinate is urro . 43429 &c. the diflance of that point from 
 the other afymptote being greater tlian the femi-axis but equal to 1 %. 
 But if the origin of the abfcilTes is taken equidiflant from the fummit 
 of the hyperbola and die interfeclion of the afymptotes, the afymptotes 
 of tlie hyperbola, whofe areas reprefent the common logarithms, are in- 
 clined to each other about 25°. 44', of which the fine is 11 = 0.43429 Sec. || 
 
 The formuliC X and Y havr alfo been deduced from the logarlth- 
 jnic § — a curve whofe abfciflcs are the logarithms of its ordinates or 
 
 converfely 
 
 • Wallirii Opera, vol. 3. p. C34 and fcq. cited by Ilutton. f Montucla. :|: Appendix, 
 
 Hyperbola. || Hutton's Malh.,Tab. J Encyclopedic au mot Logarilhmique. Appcudijc, Lo- 
 garitlmiic.
 
 78 L I F E, W R 1 T I N G S, A N D 
 
 converfely *. This curve is faid to have been invented by Edmund 
 Gunter f ; but its properties, fome of which are very remarkable, do 
 not feem to have been much known and attended to till the time of 
 Huvgens, who cn\imerates them in his Caufa gravitatls. It v? as confi- 
 dered afterwards by Leibnitz, Bernoulli, I'Hopital, and a great many 
 others. The manner in which it is treated by John Keill in the trad: 
 on the logarithms fubjoined to his edition of Euclid, facilitates very 
 much the conception of thefe numbers. In the Appendix the reader 
 "will find this curve treated in a new manner, with an enumeration of 
 fome new properties. 
 
 The fame formula X and Y are eafily deduced by the fluxicnary me- 
 thod from Neper's generation of the logarithms. From what is faid ia a 
 foregoing fedlion it is evident that (Fig. XI.) the velocity of the firft 
 moveable at the point N is to its velocity at the point N' as CN is to 
 CN'; but the velocity of the firfl moveable at the point N is the fame with 
 the velocity of the fecond moveable point at (any point of Ka) L': 
 
 therefore, in the language of fluxions, if NN'=:^, Log. (i +^) : a :: i : 
 
 . , * a * * * 
 
 1 + a, therefore Log. ( i + <^) = = « — ^'c +a^a — &c. therefore Log. 
 
 r +a — a — %^ + °^ — 8<c. If the points n' and 1' are taken, it may be 
 fliewn in the fame manner that Log. [i — «]= — a — ?'=y' — &c. 
 
 In the year 1695, Edmund Halley greatly improved the theory of the 
 logarithms, by deriving the feriefes for their conftruiflion from the prin- 
 ciple* 
 
 • Appendix, Logarithmie. f Montuda.
 
 INVENTIONS OF NAPIER. S^ 
 
 them, form the logaritlim of the number arifmg from the junction of 
 the digit at top or bottom to the figures in the firft column, corref- 
 ponding to faid four figures. When the laft of the three firft figures 
 of a logarithm, correfponding to a number formed by figures in die 
 firft column and a fignihcant digit at top, is found augmented by uni- 
 ty, thefe three figures, together with the correfpondent fours, arc moved 
 a line downwards ; by this means one avoids the miftaking one three 
 figures for another, which, without fpecial care, muft often be the cafe 
 in ufing Sherwin's, Gardiner's or Hutton's Tables. The laft column 
 contains the differences of the confecutive logarithms, together with the 
 proportional parts correfponding to the nine digits. With thefe pro- 
 portional parts one can compute by the eye alone the logarithms, not 
 in the table, of all the numbers lefs than 1029600, and, with very little 
 trouble more, thofc of all numbers lels than 10296000, as exactly as eight 
 places of figures can exhibit them. In the table of the logarithmic 
 fines and tangents, the degrees and minutes are difpofed nearly in the 
 fame manner as in Napier's Table. Each page contains eleven columns. 
 In the firft and laft are the minutes. In tlie fecond and laft but one 
 are the feconds o, 10, 20, 30, 40, 50, o, and o, 50, 40, 30, 20, 10, o, of 
 which the firft and laft zeros are in the fame line with and the reft be- 
 tween each fuccceding mhiutc. In the third, fifth, feventh and ninth 
 columns are the logarthmic fines or cofines, cofines or fines, tangents or 
 cotangents, and cotangents or tangents, according as they refer to the 
 degrees at' top and the minutes and feconds in the firft and fecond co- 
 lumn, or to the dcgreees at the bottom and the minutes and feconds in 
 the lalt penult columns. The other three columns contain the diU'cr- 
 cnces of thefe logarithms. The above defcription will become perfect- 
 ly intelligible by infpecting the following fpecimcns of tlicfc Tables. 
 
 y Tab.
 
 «4 
 
 LIFE, WRITINGS, and 
 
 TAB. DES LOG. DES NOME. NAT. 
 
 N. 14R00 L. 270 
 
 N 
 
 J 490 
 
 N 
 
 JioS 
 
 J70.2617 
 
 .Si 
 
 555' 
 
 H2 
 
 8482 
 
 
 171. 
 
 ^3 
 
 1412 
 
 i-U 
 
 4339 
 
 1483 
 
 7205 
 
 86 
 
 172.0188 
 
 87 
 
 3110 
 
 88 
 
 6029 
 
 89 
 
 8947 
 
 '73- 
 
 1863 
 
 1 
 
 2 
 
 2911 
 
 5844 
 
 ^775 
 
 3"4 
 6137 
 9.68 
 
 1704 
 4^'32 
 7557 
 
 1997 
 
 4'J24 
 7849 
 
 Oh 80 
 
 0:73 
 
 6j2. 
 
 3<'94 
 6613 
 
 9-39 
 
 ^5io 
 
 2.54 
 
 2446 
 
 I 
 
 2 
 
 3497 
 6-130 
 
 2290 
 5217 
 8142 
 
 1065 
 3,86 
 6905 
 
 9822 
 
 2737 
 
 4 
 
 5 
 
 6 
 
 7 
 
 8 
 
 9 
 
 37y' 
 6723 
 
 4084 
 70 r 7 
 
 4377 
 73'o 
 
 4670 
 7603 
 
 4964 
 781,6 
 
 5-57 
 818; 
 
 9654 
 
 2583 
 5509 
 
 >'434^ 
 
 9947 
 
 2S76 
 5S02 
 8727 
 
 0240 
 3 '68 
 6095 
 
 o53< 
 3461 
 6387 
 93" 
 
 0826 
 
 3754 
 6680 
 9604 
 
 I (K; 
 
 4046 
 
 ^972 
 9S96 
 
 1357 
 4278 
 
 :'y7 
 
 1649 
 
 4570 
 7488 
 
 9019 
 19 + 1 
 4S62 
 7780 
 
 2233 
 5'54 
 8072 
 
 2526 
 5446 
 8364 
 
 2818 
 
 5737 
 8655 
 
 OIl:{ 
 
 0405 
 
 0697 
 
 Ory8» 
 
 1280 
 
 1571 
 
 3028 
 
 • 
 
 3320 
 
 3''i ' 
 
 3903 
 
 4194 
 
 • 
 
 41^5 
 
 4 
 
 5 
 
 6 
 
 7 
 
 8 
 
 9 
 
 diff. part. 
 
 .^ 
 
 ■47 
 
 
 
 (J 
 
 .76 
 
 
 7 
 
 205 
 
 
 H 
 
 ■iS 
 
 '93 
 
 V 
 
 '05 
 
 I 
 
 ZJ 
 
 
 2 
 
 j» 
 
 
 3 
 
 U 
 
 - 
 
 4 
 
 iW 
 
 
 .■i 
 
 Mr 
 
 
 6 
 
 176 
 
 
 7 
 
 20J 
 
 19a 
 
 i 
 
 »J4 
 
 
 '9 
 
 9 
 
 ><>4 
 
 
 !« 
 
 
 
 iiii 
 
 
 
 iir 
 
 
 
 m6- 
 
 
 
 TAB. DES LOG. DES SIN. ET TANG. 
 12 DEC. 
 
 I 
 
 ff 
 
 20 
 
 
 
 
 10 
 
 
 20 
 
 
 30 
 
 
 40' 
 
 
 50 
 
 21 
 
 
 
 
 10 
 
 
 20 
 
 » 
 
 * J 
 
 • 
 
 
 fin. 
 
 9.32959.88 
 
 9.32969.50 
 9.329-9. 13 
 9.329S8.75 
 
 9.32998.38 
 9.33008. 00 
 9.33017.61 
 
 9.33027.23 
 9.33036.84 
 
 dm: 
 
 962 
 
 963 
 962 
 
 963 
 962 
 961 
 
 962 
 
 961 
 
 co-fm. 
 
 9.98985.97 
 
 9.98985.51 
 9.98985. 04 
 9.98984.58 
 
 9.9R984 
 9.98983 
 9.98983, 20 
 
 , t2 
 
 ,66 
 
 9.98982. 74 
 9.98982.28 
 
 diff. 
 
 46 
 
 47 
 46 
 
 46 
 
 ..16 
 46 
 
 46 
 
 46 
 
 , I " 
 
 CO-flll. 
 
 diff. 
 
 Cn. 
 
 diff. 
 
 9-33973-9' 
 
 9.33984.00 
 
 9- 33994- C9 
 9.34004. 17 
 
 9. •54014. 25 
 
 9-34024-33 
 9.34034.41 
 
 9.34044.49 
 9.34054.56 
 
 1009 
 
 1009 
 
 1008 
 
 1008 
 
 1008 
 icoS 
 
 icc8 
 
 1007 
 
 CO- tang. 
 
 diff. 
 
 co-laiig. 
 
 II 
 
 0.66026. 09 
 
 
 
 0.66016.00 
 
 <;< 
 
 0.66005. 91 
 
 40 
 
 c. 65995. 83 
 
 30 
 
 o.fi5985.*75 
 
 20 
 
 0.65975.67 
 
 10 
 
 0.65965.59 
 
 
 
 0.65955.51 
 
 50 
 
 0.65945.44 
 
 40 
 
 tang. 
 
 
 I 
 
 48 
 
 39 
 
 77 DEC. 
 
 TilESB
 
 INVENTIONS OF NAPIER. S5 
 
 These Tables, which are executed with a new and elegant type on 
 good paper, form a fmall odlavo volume. There is every probability 
 in favors of their corre<5lnefs. They are copied from the London edi- 
 tion of Gardiner printed in 1742, which is in the hlghcft cftimation 
 for that quality. Ivlcflieurs Callet. Leveque and Prud'homme, three 
 good mathematicians, revifed the proof Iheets, as did alfo the editor M. 
 Jombert three feveral times. M. Didot fenr. the printer formed the 
 models of the types and founded them on purpofe, and the editor avers 
 that, during tlic courfe of the imprefhon, none of the figures came out 
 of their place; a precious advantage which he imputes to the juflnefs 
 of the principles that M. Didot has eflabliflied in his foundery. 
 
 There is an additional improvement, which I am furprlfed none of 
 the editors of our common logarithms has thought of making. What 
 I allude to is the uniting, to the tables of the logarithms of the natural 
 nximbers and of the fines and cofmes, the logarithms of their recipro- 
 cals (their aritlimetical complements *, as they are called). By this 
 means, all the common operations by logarithms might be performed by 
 addition only, without any trouble. The logarithms of the natural 
 numbers might be dlfpofcd on the left hand, and thofe of their recipro- 
 cals on the right hand pages. The charafteriflics of the latter, being 
 equal to the difference between 10 and the number of integral figures iii 
 the natural numbers, woiild be as cafily found as thofe of the former. 
 The logarithms of the reciprocals of the fines and cofmes might, in each 
 page, be put in the fame line with the logarithms of the fines and cofines, 
 
 having* 
 
 * ITie arithmetical complements of iLc logaritlims were Gift thought of by John Spccdcll, who,- in 
 Lis //."ui logarilbmt firft puLlilVicJ in 1^)19, and feveral times aftcnvards, awijtd the incoDvcuicocc 
 •f the figns in Napicri logarithms by that contrivance.
 
 S6 LIFE, WRITINGS, and 
 
 having their common differences between them, as the logarithms of the 
 tangents and cotangents, wliich arc reciprocals of each other, have 
 theirs. It is very likely that the prefcnt edition of the Tables portativcs 
 will foon be exhaufted. If, in a fecond edition, M. Jombert adopts the 
 propofed amelioration, he will do an effcntial fervice to the communi- 
 ty. I . The computation might be accompliflied, by a good arithmetici- 
 an, in little more than [three hours labour every day for half a year. 
 2. The type and length of the page being the fame, the book would be 
 little more than a fourth part thicker, and would flill be of a convenient 
 fue. 
 
 In the month of May, 17S4, there were publiflied propofals fpr 
 publifliing, by fvibfcription, A 'Table of Logarithmic fines and tangents^ ta 'en 
 at fight to every fecond of the quadrant^ accurately computed to feven places off- 
 gures befides the index : to ivblcb ivill be prefixed a table of the logarithms of 
 numbers from i to 1 00000, infcribcd^ by permiffion^ to the right honourable and 
 honourable the Comm'ffioners of the Board of Longitude^ by Michael Taylor^ one 
 of the computers of the Nautical Ephemeris, and author of a Sexage/imal Table ^ 
 publ'ffjed by order of the Commiffioners of the Board of Longitude. The plan of 
 this work was fvibmitted to the Board of Longitude, who came to a re- 
 folution to give Mr Taylor a gratuity of three hundred pounds flerling 
 towards defraying the expence of printing and publifliing it. This cir- 
 cumftance ought to be a fufficient recommendation of Mr Taylor, and 
 it is to be hoped, that his laborious and ufeful undertaking will meet 
 with the encouyagement and recompence from the public which it fo 
 Juflly deferves. In the fpecimen annexed to the propofals, the degrees 
 being as ufual at the top and bottom of the page, tlie feconds occupy 
 
 the
 
 INVENTIONS OF NAPIER. 79 
 
 ciples of common Algebra independently of any curve. He was tlie 
 firfl alfo, if I n:iillake not, that gave the general feries for computing 
 the numbers correfponding to given logarithms *. The analytical the- 
 ory of logarithms, in the Appendix, is nearly on Halley's plan, but 
 was materially finiflicd before the author faw his treatife. 
 
 To defcribe, or enumerate, all the tables of logarithms, which have 
 been publiflied fmce the invention of thefe numbers, would be tedious 
 and ufelcfs, and indeed next to impoffiblc. We Ihall reftricl ourfelves 
 to thofe which are tlie mod confiderable and the moft ufcful. 
 
 In the year 1624, Benjamin Urfmus, mathematician to the Ele<flor 
 of Brandenburg, publiflied at Cologne, with his Tr'igoiiometria^ a Tabic 
 of Napier's logarithms of the fines to every ten feconds of tlie quadrant. 
 He feems to have been at much pains in computing it, and, in order to 
 obtain the logarithms true to the neareft unit in the eight figure, he fup- 
 pofcd the radius followed by an additional zero, as Napier had advifed f . 
 
 In the fame year, Kepler publiflied, at Marpurg, his Ch'ilias Loga- 
 r'lthmorum ad todldem nnmeros rotitndos &c. and, in the year follov.-ing, a 
 fupplement to it. \\\ this table, the logarithms are of the fame kind 
 with thofe of Napier, but adapted to fines in arithmetical progreflion. 
 
 Small tables of the fame fpecles of logarithms have been publiflied 
 by T. Simfon in his fluxions, by Dr Hutton in his Math. Tab. and by 
 a great many others, to eight places. In Eulcr's Introdu^io in aiuljfut 
 
 X hifiiiUorum 
 
 • Phil. Trunf. for 1695. 1 Krpl. Epift>
 
 8o L I F E, W R I T I N G S, A N »' 
 
 iiifimtoriim^ there is a Imall table of the firft ten natural numbers witH. 
 their logaritlims to twenty fix places ; and, in Bertrand's work formerly 
 mentioned, there are the logarithms of a great many of the firfl hun- 
 dred, natural numbers, and of feveral others, to the fame number of pla- 
 ces. Some of thefe differ from the truth, by fome units only, in the laft 
 figure, and the logarithm of 6 1 is wrong in the fixtecnth figure from 
 the left hand. In the Appendix there is a table of Napier's logarithms 
 of the firft hundred and. one natural numbers to twenty fevcn places. - 
 
 In the year 1624, Briggs publKhed at London his Aritbmetka Logcr- 
 rhhmica. This work contains Briggs' or the common logarithms, and 
 their differences, of all the natural numbers from i to 20000, and from 
 90000 to 1 00000 to fifteen places, including the index or characleriflic. 
 In fome copies, of which there is one in the Library of the Univerfity 
 of Edinburgh, there is added the logarithms of the numbers from 
 1 00000 to loiooo, which Briggs had computed after the former had 
 been printed off. Before his death, which happened in 1630, this au* 
 thor completed alfo a table of the logarithmic fines and tangents to fif- 
 teen places, for the hundredth" part of every degree of the quadrant, and 
 joined with it the natural fines, tangents, and fecants, which he had be- 
 fore calculated. This work which Briggs had committed to the care 
 of Henry Gellibrand, at that time profeffor of aflronomy in Grefham 
 College, was tranfmitted to Gouda, where it was printed under the in- 
 fpe(flion of Ulacq, and was publifhed at London in 1633, with the title 
 of Trigonometr'ia Br'ilann'ica. 
 
 ThesB" tables of Briggs' have not been equalled, for their extenfive- 
 ncfs and accuracy together ; thofe of his' logaritlims that have been re- 
 examined
 
 INVENTIONS OF NAPIER. tt 
 
 examined having feldom been found to differ from the truth by more 
 than a few units in the fifteenth figure. 
 
 In the year 162S, Adrian Ulacq of Gouda, in Holland, after filling 
 up the gap betwixt 20000 and 90000, which Briggs had left, repub- 
 Ulhed the Aritbmctica Lagar'ithmica^ together with a table of the loga- 
 rithmic fines, tangents, and fecants, to every minute of the quadrant. 
 Some years afterwards, he publillaed his 'Trigonometria Artificiality con- 
 taining Briggs' logarithms of the firfl: twenty thoufand natural num- 
 bers, and the logarithmic fines and tangents, witli their differences for 
 every ten feconds of the quadrant. In both thefe works, the logarithms 
 are carried to the eleventh place including the index, and are held in 
 much eftiniation for their corrednefs. 
 
 Abraham Sharp, of YorkHaire, publifhed with his Geometry Improved^ 
 in 1 7 1 7, a table containing Briggs' logarithms of the firfl hiindred na- 
 tural numbers, and of all the prime numbers from 100, to 1 100 and of 
 all the numbers from 9999S0 to 1000020, to fixty two places including 
 the charaderiftic. There is the greatefl probability of all tlicfe loga- 
 rithms being correal. The laft forty-one [from 999980 to 1000020] 
 ■were verified afterwards by Gardiner. 
 
 Tables of the logarithms, carried to fo great a number of places as 
 thofe of Sharp, Briggs, and Ulacq, are feldom ufed ; tlic logarithms to 
 eight places inclufive of the charaifleriflic being fuflicient for all com- 
 mon purpofcs. The moft ufcful tables are thofe which have the loga- 
 rithms corrc(5t to the ncarcfl unit in the eight figure, difpofed fo as to 
 
 take
 
 8:^ L I F E, W R I T I N G S, AND 
 
 tAkc up little room, and, at the fame time, to afford the eaileft and moft 
 Ipeedy means of finding the intermediate logarithms, or numbers cor- 
 refponding to given numbers or logarithms The form of the ta- 
 bles bed adapted to anfwer thcfe purpofes was firfl introduced by Na- 
 thaniel Roe, a clergyman in Suffolk, in his Tabula Logarithm'ica, print- 
 ed at London in 1633. This form was improved by John Newton, in 
 his Trigotiometria BrUannlca publiflied at London in 1658, and by Sher- 
 win in his Mathematical Tables, of which the firil edition was printed iri 
 1 705. It has received additional improvements in Mr Callet'$ edition 
 of Gardiner's Tables printed at Paris in 1783. * 
 
 The difpofiton of the tables is as follows : Each page of the logarithms 
 of the natural numbers is divided into twelve columns. The firfl co- 
 lumn, titled N at top and bottom, contains the natural number. In 
 rhc fecond column, marked O, are the logarithms, without the charac- 
 teriflic, of thefe numbers ; the three firft figures, belonging to the lo- 
 garithms of more numbers than one, arc fcparated by a point from the 
 other four figures of the logarithm of the firjl of thefe numbers and 
 are left out before the other four figures of the logarithms of the reft. 
 In each line of the next nine columns, marked with the nine Cgnificant 
 digits I, 2, 3, &.C. are four figures, which, united to the firll three ifola- 
 ted figures of the fecond column in the fame line with them, or above 
 
 them, 
 
 • Tables poitatives de Logaritluncs, publices a Londrcs par Gardiner, Angmentec ct perfe<Stio- 
 nees dans kur diTpofitlon par M. Callct, et corngees avec la plus fcrupideufe exaaitude : contenant 
 ks lo-rarithmes dcs nombre depuis i jufqu'a 103960, les logarithmes des finus and tangentes, de feconde 
 en feconde pour les deux premiers degres et de 10 en 10 fecondes pour tous les degres du quart de 
 circle ; precedees d'un precis elemeiitalre fur I'explication et I'ufage des logarithmes et fur Icur ap- 
 plication aux calculs d'interets, a la Geometric-pratique, a I'Allronomie et a la NaWgation } fulvics 
 de pluTteurs tables intereflantes et d'un difcours qui en facilite I'ufage. A Paris 1783.
 
 INVENTIONS OF NAPIER. 87 
 
 cHc firil column : the minutes are difpofed along the tops and bottoms 
 of the other columns : immediately below the minutes at top Hand 
 the chara<fteri(lics, and below them the three next common figures of 
 the logarithms ; the other four figures filling the columns. It is to be 
 regretted, that an improvement^ fimilar to M. Calkt's, has not been 
 adopted in this work, the printing of which was begun before the date 
 of the propofals. 
 
 The tables of logarithms which, with thofe that have been men- 
 tioned, are moft in eftimation, are thofe of the edition of Sherwin, 
 which was corrcdled and publiilied by Gardiner in the fame year (1742) 
 with his own tables — Thofe by Deparcieux *, and thofe of the fmall 
 editions of Ulacq publifhed at Lyons in 1670, and lyoof. 
 
 The London edition of Gardiner, which has been defervedly efteemed 
 as containing the mofl accurate fet of tables, is not entirely free from er- 
 rors. There is, at the end of Dr Hutton's tables, a Hft of about fifty er- 
 rors in tlie logarithms of the natural numbers, fines and tangents; twenty 
 of which he himfclf difcovered in collating the proofs of his book with 
 the like parts of Gardiner's; all of thefe, however, that gentleman ob- 
 ferves, are not in all the copies of this edition. In the Avignon edition 
 of Gardiner (1770), the errors pointed out by Dr Hutton are above 
 fevcnty. All the errors of the London edition are correded in the 
 Tables portatives, excepting that of the logaiithm of the natural numbers 
 
 64445- 
 
 Z Befori 
 
 * MontucU^ f Hutton.
 
 88 L r F E, W R I T I N G S, g<e. 
 
 Before concluding this fedtion, we fliall fay a few words of the lo- 
 garithms called logiftic. The logiflic logarithm of a number of fe- 
 conds is the excefs of the logarithm of 3600" above the logarithm of 
 tliat number of feconds. A table of thefe logarithms was firfl given 
 by Strut in his Ajlronomia Carolma publiflied in 1661 *. A fimilar one 
 is given in feveral of the common logarithmic tables. 
 
 SECTION 
 
 * Tab. portatlves.
 
 SECTION VII. 
 
 THE USE OF THE LOGARITHMS. 
 
 1 HE general ufe of the logarithms, as was before obferved, is to con- 
 vert every fpecies of multiplication and divifion into addition and fub- 
 tra<flion, and to raife quantities to any given power, and to extract their 
 roots by eafy multiplications and divifions. Examples of thefe operati- 
 ons, particularly in trigonometry, arc prefixed to almoil all the moft con- 
 fiderable tables of logarithms. We beg leave to refer the reader to 
 Gardiner, Callet, Sherwin, and Hutton, where he will find the theory, 
 conllrudtion, and application of thefe numbers. 
 
 The theory of the logarithms has put it in our power to fol^c, with 
 great eafe, an equation in algebra, which before could not be folved but 
 with difficulty and tatonnement. In the equation a^ z=.h, li bis the un- 
 known quantity, its value is found by multiplying a by itfelf as often 
 as there arc units in x — i : Again, if a is the unknown quantity, its 
 value may be found by extradling the *th root of b. But if x is the 
 unknown quantity, algebra, without the logarithms, can furnilh no di- 
 Te(5l rule for finding its value. This, however, is eafily accomplilhed 
 
 by
 
 90 L I F E, W R I T I N G S, A N D 
 
 by the afliftancc of the logarithms. Let L denote the logarithm of the 
 tjuantity to which it is prefixed. Now fince z.'^-zzb^ it is evident tliat 
 
 Lb 
 
 La = '*Lb; but La'^izxLa: therefore xLa= Lb : therefore x: 
 
 La* 
 
 The Solution of Equations of the form a^=:b is of great importance 
 in political arithmetic. Suppol'e that a quantity at firft »;, being in- 
 creafed at the end of every equal portion of time by a quantity c, aug- 
 ments at the rate r; and that it is found, at the end of a number x of 
 thefe portions of time, to be augmented to « ; the equation exprefling 
 the relation of thefe quantities to each other is ( i + rY = « + — ■ 
 
 m-\ 
 
 r 
 
 By the help of the logarithms, this formula, among other purpofes, 
 ferves for finding with facility in what time a fum of money n might 
 be paid oflT by finking at firft a fum w, and at the end of every year 
 another fum c, leaving their intereft r to accumulate. In what time, 
 for example, might the national debt of Great Britain, 270 millions of 
 pounds Sterling, be extinguiihed by finking one million every year and 
 allowing its intereft, five per cent per annum, to accumulate ? The cal- 
 culation is as follows.. 
 
 c 
 w=:27ooooooo n-\ — = 290000000 Log. = 8 . 4623980 
 
 c 
 mz= I 000 000 m-\ — = 21000000 Log. = 7 . 3222193 
 
 c 
 
 c= 1000000 Log. / «-j — \ =1.1401787 
 
 
 rzz
 
 INVENTIONS OF NAPIER. 91 
 
 I '•I 
 i4-r= I -I ='—Log. 2im . 32221Q3 
 
 r= 
 
 100 
 
 I 
 
 "20 
 
 r 
 
 = 20 000 000 
 
 Log. 20=: I . 3010300 
 
 Log. (1+/) =0.0211893 
 
 _ r . 1 40 1 787 _ 1 14017S7 
 "0.0211893" 211893 
 
 Log. 11401787 = 7.0569729 
 Log. 211893=5.3261167 
 
 Log. .v= I . 7308562, x = s^ . 809 
 
 In lefs than fifty four years, therefore, the Britifh nation might get 
 quit of their debt, if they could raife annually a million Sterling, over 
 and above the amount of the intereft of that debt and the expences of 
 government. 
 
 The fame equation under the form 
 
 fi = {m+—)x{i+rY- 
 
 c 
 
 r r 
 
 fcrves for computing the number ?/ of inhabitants of a country which, 
 having at firfl: in inhabitants, has received every year for x years a 
 number c of foreigners, and has increafed annually at the rate r. For 
 example, fuppofe the number of the inhabitants of the United States of 
 North America to be at prefent three milUons, that they receive ten 
 thoxifand emigrants yearly, and that the population in that country in- 
 creafes at the rate of one to twenty per annum j What will be the num- 
 
 A a ber
 
 <ja L I F E, W R I T I N G S, A N D 
 
 ber of inhabitants of thofe States a hundred years hence ? The calcula- 
 tion is as follows : 
 
 01 = 3000 000 w+ — = 3200 000 Log. = 6 . 5051500 
 
 f= 10000 looLog. (i-t-r) = Log.(i-|-r)'°°=2. 1189300 
 
 rzz.- 
 
 Log.(»z+— ) (i+r)' =8. 6240800 
 
 20 ^ r 
 
 — = 200000 [rn-\ ) (i-j-r)* =420800000 
 
 r r 
 
 c 
 X =: 100 — ^= 200000 
 
 r 
 
 7^ = 420600000 
 
 Hence it appears, that were tlie lands of the United States extenfive 
 enough, and were the fame circumflances, favourable to population as 
 at prefent, to continue for one hundred years, the number of their in- 
 habitants would amount to more than four hundred and twenty mil- 
 lions, which is a good deal greater tlian twice the number of inhabi- 
 tants computed to be in all Europe. 
 
 The logarithms alfo, after the invention of fluxions, give rife to a 
 fpecies of calculus called the exponential. This calculus was invented 
 by John Bernoulli and firft publiflaed in the year 1697 *• ^^ ^^ found- 
 ed on tliefc two principles : i. The logarithm of the pov/er of a quanti- 
 ty is equal to the produdl of its exponent by the logarithm of its root, 
 or xLa = La*: 2. The fluxion of the logaritlun of a quantity is pro- 
 portional to the quotient of tlie fluxion of that quantity by that quan- 
 tity 
 
 * De Serie, Infin. Jacobi Bernoulli.
 
 INVENTIONS OF NAPIER. ^3 
 
 a 
 
 city or La = — . The exponential calculus is neceffary for the inveftiga- 
 
 tion of curves, the exponents of whofe abfciffes and ordinates, or their 
 functions in the equations to thefe curves, are themfelves variable quan- 
 tities, tf, V, ss, &c. Exponential curves, fuch, for example, as have for 
 
 V V 
 
 the value of their ordinates x", x" ^ x' , &c. are faid to be of the firil, 
 fccond, third. Sec. order. What are the fubtangcnts, curvatures, ar- 
 eas, &c. of curves of this nature ? Let SMM', (Fig. XII.) be any curve, 
 fc its abfcis CSP'r=A; and ordinate P'M' = y and let there be another curve 
 (r^.|M,' having the fame abfcifs with the former, and its ordinate VM'fj. = 
 zrrx^ for example, let/V/ be an ordinate infinitely near to p'y.' and 
 (jt/v perpendicular to it, and let r'fjJ be a tangent at the point yJ : the 
 'limilar ti'iangles r'p'f/J and ^/^'/ give t'P' :p' ^' : : jW-V ; »V', therefore the 
 
 7 X * 
 
 fubtangent r'p'— '.' : but zr^xy therefore LzzrLx>'=:yLx therefore Lz 
 
 z 
 
 • • 
 
 _^— — z • * vx * 
 
 = yLx that is — = yLx+ y Lx = — -|-yLx, and therefore 7'p' — 
 
 X X 
 
 — . Hence it is evident that the relation of x to y, that is, 
 y X -|- ^y Lx 
 
 the Equation to the curve SMM' being given, the fluxion of y may be 
 exprcfl'ed by fome fundlion of x, and Its fluxion may be obtained; which 
 
 X x 
 
 value of the fluxion of y being fubfliituted in the fradllon — : '- — ; 
 
 y X -f- X y Lx 
 and the fluxion of x expunged from its numerator and denominator, 
 there will be obtained a finite cxprcflion of the fubtangent r'p' of the 
 
 curve
 
 94 LIFE, WRITINGS, and 
 
 curve c-y.u.'. For example, lee the cui"ve SMM' be tlie logarltlinilc : we 
 
 ' - r " . - 
 
 have yzr Lx * : therefore y = — ; therefore r 'p' =—;:^ — • From the 
 
 'V art X^^ 
 
 value of the fubtangenc and from the equation (z z=x"^ to the curve o-jj^jjiJ 
 
 a great many of its properties are eajQIy deduced. The ordinate Sir at the 
 
 fummit of the curve is equal to the abcifs CS : for y=Lx=o and z^^x" 
 
 =::K^S. The tangent at the point <r is parallel to the axis CSD : for Lx 
 
 x° 
 =o and Tj6'= — =:*oo. The ordinate a is an afymptote to tlic rurve 
 
 acrtn : for x=o and Lx= — oo and therefore t'p'^=.— ==o. The tan- 
 
 ■^ 200 
 
 gent palling through the point c meets the curve (t^m.,!/,' at the extremity 
 of the ordinate z=^^/x : for x=t'^'=^— ; therefore Lx=^ The tan- 
 
 2 J~jX 
 
 gents to the points M and jo,, where y=—;= and z=::xv^', meet in the 
 fame point r in the axis : For the fubtangent of the logarithmic <: is = 
 
 X I 
 
 xLx =rr/>=— --; therefore L*x=i and Lx::=-y=i The ciirve <rf/.(jJ may 
 
 be calkd the Numer'ico-Logarithmk : and if the equation vv^ere (Lx)'':==z 
 or y'^:=z there wo\ild be generated a curve which might be called the 
 Logdrithmo-numeric. 
 
 The above fmall fpecimen may fufBce for giving an idea of the ufe 
 Cf the exponential calculus. The reader will have obferved that we 
 have made ufe of Napier's, or, as they have been called, the natural 
 logarithms. It wovild have been an eafy aflair to have made ufe of any 
 
 other 
 
 * See AppcndiXi
 
 INVENTIONS OF NAPIER. 95 
 
 other logarithms. It may here be obferved that the logarithmic itfelf, 
 is an exponential curve of the fir ft degree or order : for the abfcifs x is 
 of the form cJ'y c being a conftant quantity greater than unity and ha- 
 ving I for its logarithm. 
 
 Those, vi^ho vi'ifli to enter fully into this fubjecl, may confult the 
 Works of John Bernoulli, and the Analyfe des Itifinime/ts pet'its of the Mar- 
 quis de I'Hopital with M. Varignon's EclairciJJ'cmcnts. 
 
 Another ufe of the logarithms is to folve the problems of failing 
 according to the true chart, independant of a table of meridional parts. 
 It was firft publifj-ied, by Mr H. Bond, about the year 1645, that the 
 meridian line was analogous to afcale of logarithmic tangents of half the com- 
 plements of the longitudes *. Nicolas Mercator feems to have been the firft 
 to demonftrate this property of the meridional line. But he kept his 
 demonftration fecret. James Gregory firft publifhed a demonftration 
 of it in his Exercltatlones Geometrlca. Halley, afterwards, (about the 
 year 1695) gave a much better one in the philofophical tranfic- 
 tions. On this fubjecl the reader may confult Robertfon's Navigation, 
 where he will find it treated in a plain manner and illuftrated witli ex- 
 amples. 
 
 The logarithms alfo exhibit the alTymptotic areas of the hyperbola f. 
 
 They are likewifc of great fervice for the fummation ot" infinite fe- 
 ricfes in the calculus of fluents. This is true particularly of Napier's 
 
 B b logarithms. 
 
 • Pliil. Trans. N02 191 f Sec Scft, vi. and Appcndiii
 
 95 L I F E, \V R I T I N G S, See. 
 
 logarithms. The fum, for example, of about feven hundred millions 
 of terms of the infinite feries i — T+r — i+> ^c* ^' equal to 0.69314 
 718, Napier's logarithm of the number 2. 
 
 SECTION
 
 SECTION VIII. 
 Napier's improvements in the theory of trigonometry. 
 
 VV E obferved before that the Arabs, fetting afide the chords of the 
 double arcs, which rendered Trigonometry very complicated among 
 the ancients, made ufe of the halves of thefe chords to which they gave 
 the name of the Sinus. To that ingenious people \\c owe alfb the three 
 theorems which are the foundation of our modern fpherical trigonome- 
 try. By thefe theorems all the cafes of rc(fl:angular fpherical triangles 
 and all the cafes of oblique fpherical triangles may be refolved, except- 
 ing when the three fides, or the three angles only, are the data. It was 
 Regiomontanus who firfl invented two theorems for the folution of 
 thefe two cafes : by which means the theory of trigonometry was per- 
 fccfhed. One of thefe theorems which ferves for finding an angle fi-om 
 the three fides is, The reSlangle under theftnes of the two fides of any fpheri- 
 cal triangle is to the fqnare of the radius ; as the deference of the verfcd fines 
 of the bafe and the difference of the two fides is to the verfcd fine of the vertical 
 angle. The other theorem, of itfelf, is not fufficient for the purpofe of 
 finding a fide from the three angles. 
 
 This
 
 98 L 1 r E, WRITINGS, and 
 
 This lafl cafe, however, may be refolved into the former by means ot 
 the fupplemental triangle, fo called becaufe its fides are the fupplements 
 of the angles of tlie other. This invention is due to Bartholomus Pi- 
 tifcus *, who liourifhcd in the beginning of the feventeenth century. 
 
 The improvements made by Napier on this fubjecfl are chiefly 
 three. I. The general rule for tlic folution of all the cafes of rectangu- 
 lar fpherical triangles, and of all the cafes of oblique fpherical triangles, 
 excepting the two formerly mentioned. 2. A fundamental theorem by 
 wliich the fcgments of the bafe, formed by a perpendicular drawn from 
 the vertical angle, may be found, the three fides being given. This, 
 with the foregoing and the property of the fupplemental triangle, fervcs 
 for the folution of all the cafes of fpherical triangles. 3. Two propor- 
 tions for finding by one operation both the extremes, the three middle 
 of five contiguous parts of a fpherical triangle being given. 
 
 These theorems arc announced by Napier in terms to the following 
 import : 
 
 I. Of the circular parts of a rectangular or quadrantal fpherical tri- 
 angle. The 7-eEl angle under the radius and the fine of the middle part is equal 
 to the rectangle under the tatigcnts of the adjacent parts and to the reElanglc un- 
 der the cofines of the oppofitc parts. The right angle or quadrant fide be- 
 ing negle(fted, the two fides and the complements of the other three 
 natural parts are called the circular parts ; as they follow each other as 
 
 it 
 
 * Pitifco aliquid tribuo in fUTsAan arcuum in angulos, et vicifliir.. Kep. Epift. 29J.
 
 'INVENTIONS OF NAPIER. 99 
 
 it were in a circular order. Of thefe any one being fixed upon as die 
 middle part, thofe next to it are the adjacent, and thofe farthefl: from 
 it, the oppofite parts. 
 
 2. "Tbe reS! angle under the tangents of half the fum and half the difference 
 of the fegments formed at the bafe by a perpendicular drawn to it from the 
 vertical angle of any fphcrical triangle^ is equal to the re^angle under the tan^ 
 gents of half the fum and half the dfference of the two fides. 
 
 3. 'The fines of half the fum and half the dfference of the angles at the bafe 
 of any fpherical triangle are proportional to the tangents of the half bafe and 
 half the dfference of the fides. 
 
 4. The cofines of half the fum and half the dfference of the angles of the 
 bafe of Mn\ fpherical triangle^ are proportional to the tangents of half the bafe 
 and half the fum of the fides. 
 
 Napier gives alfo the two following theorems for finding an angle, 
 the three fides of any fpherical triangle being given. 
 
 5. The rc^ angle under the fines of the two fides is to the re^ angle under 
 the fines of half the fan and half the dfference of the bafe and the difference of 
 the two fides, as thefquare of the radius is to the fquare of the fine of half the 
 vertical angle.' 
 
 C). The rectangle under the fines of the two fides is to the rectangle under 
 ihefncs of half the fum and half difference of the fum of the two fides and the 
 bafe, as thefquare of the radius is to thefquare of the cofme of the vertical angle, 
 
 C c Fox
 
 loo L I F E, W R I T I N G S, A N D 
 
 For the demon ftration of the vartous cafes of the firft of thefe fix 
 propolltions, he refers to the elementary books on trigonometry then 
 in ufe. This propofition is not fo fufccptible of a dire<5l demonlbatlon. 
 The dcmonflration perhaps the neareft to a dired one is given in the 
 appendix ; of -which dcmonflration the liint is taken from Napier. 
 
 His demonftration of the fecond propofition is extremely elegant and 
 of an uncommon call. The reader on thefe accounts, it is prefumed, 
 will be very glad toj'cc the fubllance of it ; which is as follows : 
 
 Let a plane NJN (Fig. XIII.) touch the fphere ADP at the point A, 
 the extremity of its diameter PA. Upon the furface of the fphere lee 
 there be dcfcribed the triangle AXy acute in 7, or Ax^ obtufe in S^ 
 Let the fme A>. and the bafe Ay or AS be produced to the point P. 
 With the pole X and dirtance Xy or its equal xS let the fmall circle of 
 the fphere Cyu interfedling xP in e and xA in s be defcribed : and' 
 from X let the arc X^> be drawn perpendicular to ACy. Ay is the fum 
 of the fegments of the bafe and A^ their difference. Ae is the fum of 
 the fides and At their difference. Let there be fuppofed a luminous 
 point in P : The Ihadows, A, b, and c, of the points A, S and 7, upon 
 the plane MN, are in the fame flraight line, becaufe the points A, ^, 
 7 and P are in the fame circular plane : alfo the fhadow A, d and e, of 
 A, i and e, upon the plane MN, are in the fame fcraight line, becaufe 
 A, J, s and P are in the fame circular plane. Since PA is perpendicular 
 to the plane MN, the plane triangles PAc, PAb, PAe and PAd are red- 
 angular in A : therefore, to the radius PA, the flraight lines Ac, Ab, 
 Ae and Ad, are the tangents of the angles APc or AP7, APb or AFC, 
 
 APc
 
 INVENTIONS OF NAPIER. loi 
 
 APe or Ap6 and APd or APj rcfpeclively. But thefe angles, being at the 
 circumference of the fphere, have for their meafures the halves of the 
 arcs intercepted by their fides : therefore Ac, Ab, Ae and Ad are the tan- 
 gents of die halves of A7, Ab, As and At refpedively. Now (by optics) 
 the fliadow of any circle, defcribed on the furface of the fphere, pro- 
 duced by rays from a luminous point fituated in any point of that fur- 
 face excepting the circumference of the circle, forms a circle on the 
 plane perpendicular to the diameter at whofe extremity the luminous 
 point is placed : therefore the points c, b, e and d are in the circumfe- 
 rence of a circle : therefore Ac X Abr=Ae X Ad. (^E. D. 
 
 The third and fourth propofitions are not dcmonflrated by Napier. 
 He probably deduced them from the fecond in a manner fimilar to 
 that in the appendix ; where the reader will find all of thefe and fome 
 other theorems of the fiime kind, demonilrated. Napier had left the 
 third propofition under a clumfy form. It was put into the form above 
 given by Briggs in his Lucubmtmtes annexed to the Canoms Mirifici Con- 
 f.ru£lio. This circumftance is not the fule mark of this work being a 
 p/ofthumous publication. 
 
 The fifth propofition is deduced by Napier from the theorem of Re- 
 giomontanus, and it is likely he derived the fixth from tlic fame Iburce. 
 To thefe two theorems the logarithms are much more applicable than 
 to that of Regiomontanus. 
 
 Since Napier's time the chief improvement made in the theory of- 
 tiigonometry is the application or the calculus of lluxions to it; for 
 >vluch we are indebted to Cotes.. 
 
 M..
 
 102 LIFE, WRITINGS, and 
 
 M. PiNGRE, in the Mcmoires de mathematique et ds pbyfique for the year 
 175C, reduces the folutloa of all the cafes of fpherical triangles to four 
 analogies. Thcfe four analogies are in fad, under another form, Napi- 
 er's Rule of the circular parts and his fccond or fundamental theorem, 
 ■with its application to the fupplemental triangle. Although it would 
 be no dliljcult matter to get by heart the four analogies of M. Pingre, 
 yet there arc few bleiled with a memory capable of retaining them for 
 any confiderable time. For this reafbn, the rule for the circular parts, 
 ought to be kept- under its prefent form. If the reader attends to the 
 circumflance of the fccond letters of the words tangents and coftnes being 
 the fame with the firft of the words adjacent and oppofite^ he will find 
 it almoft impollible to forget the riile. And the rule for the folution 
 of the two cafes of fpherical triangles, for which the former of itfelf is 
 inflifTicient, may be thus expreflcd : Of the circular parts of an ohliqiis 
 fpherical triangle^ the reti angle under the tangents of half the fum and half the 
 difference of the fegments at the middle part (formed by a perpendi- 
 cular drawn from an angle to the oppofitc fide), is equal to the reElangle 
 under the tangents of half the fum and half the difference of the oppofite parts. 
 By the circular parts of an oblique fpherical triangle are meant its three 
 fides and xhc fupplcments of its three angles. Any of thefc fix being af- 
 fumcd as a middle part, the oppofite parts are thofc two of the fame 
 denomination with it, that is, if the middle part is one of the fides, tlie 
 oppofite parts are the other two, and, if the middle part is the fupplement 
 of one of the angles, the oppofite parts are the fupplement of the other 
 two. Since every plane triangle may be confidered as defcribed on the 
 furface of a fphere of an infinite radius, thefe two rules may be applied 
 to plan? triangles, provided the middle part be reftrided to ^ftdc. 
 
 Thus
 
 INVENTIONS OF NAPIER, 103 
 
 Thus it appears that two fimple rules fuffice for the fohition of all 
 the poflible cafes of plane and fpherical triangles. Thefe rules, from 
 their neatnefs and the manner in which they are expreffed, cannot fail 
 of engraving themfelves deeply on the memory of every one who is a 
 little verfed in trigonometry. It is a circumftance worthy of notice 
 that a perfon of a very weak memory may carry the whole art of tii- 
 gonometry in his head. 
 
 D d APPENDIX.
 
 APPENDIX. 
 
 ANALYTICAL THEORY OF THE LOGARITHMS. 
 
 I. XjET the confecutive terhis of an infinite geometrical progreilion 
 differ infinitely little one from another; it is. evident that, any deter- 
 mined quantity c greater than unity being the bafis of the progrefTion, 
 there will be fome term f *=;« any given quantity. 
 
 2. The exponents of the terms of that progrefTion are faid to be the 
 logarithms of thofc terms : Thus the fymbol L denoting the logarithm 
 of the quantity to which it is prefixed, Lr*''=rt:x. Hence if c"'-=zm ; 
 then Ijm=x and L— ^ — x= — L;;/. 
 
 THEOREM T. 
 
 3. Ibe logar'ithvi of a prodiiSl is equal to the fiim of the logarithms cf its 
 factors. For fmce Y.c''= x and hc^^^z (2), it follows that he' •{-L.c-==x 
 
 +s; but .v + ^=Lr'+-' (2) :=Lc'Xr: therefore Lr'' Xr=L^*-;-Lr^ 
 Hence if c'=m and c'=/i (i) ; then Lww^^Lw-f-L" and L^^=Lot — L'/. 
 
 THEOREM.
 
 io6 A P P E N D I X. 
 
 THEOREM II. 
 
 4, The hgarithm of a power is equal to the produEl of its exponent hy 
 the logarithm of its root. For, fince Lc *=.v, it follows that }iLc''=nx ; but 
 «A=: Lf"( 2), therefore Lf'"=«Lc\ Hence if c'^^wz, then Lot"=«Lw. 
 
 PROBLEM I. 
 
 5. To exhibit the logarithm of a given number. Since c°=i, if d is an 
 infinitely fmall quantity and ^o. any finite quantity, it is evident that 
 /=i +1. Now L/^r^ (2), therefore ^=rL(i +^), therefore /V=/L(i + 
 ^)=L(i +;;)'■ (4). Let (i+^)'=i+^; we have /V=//^(i-t-^)v— />: 
 therefore, developing the furd quantity (i+^)t, making /=:oo, and re- 
 ducing 
 
 L(i+«)=^(^-4+^&e) X 
 
 Hence, if ^2 is negative, 
 
 L(i-«)=-/.(a+^+f +&C) Y 
 
 Hence, by fubtrading Y from X 
 
 6. The above formulae are of no ufe for the calculation of the loga- 
 rithms if a is fuppofed an integer. Let therefore m and n be any 
 pofitive numbers, m being greater tlian n ; and 
 
 ^mo. Let a=^, then 1+^=^, ^-a=^^ and-i±l^^, and 
 the formulas X, Y, and Z become, by fubftitution. A, B, and C.
 
 APPENDIX. 1C7 
 
 L(i^)=L(«;-;;)-L;.=-^.(^+,-^+3^.+&c) B 
 
 2^01. Let a=:~—; then i — a ~~ - " ■-, and the formula Y becomes D 
 L(-^) =hm—Um + ;/) -—u,i-^ -j-^^, +-r-~. +Scc) D 
 
 T,lio. Let fl = — ^; then i — ^zr— 4— , and the formula Y becomes E 
 
 L(-^) =L//— L(;« + ;/) -—uX-J^ + .^4_, + -^-^, + 8cc) - E 
 
 4/c. Let a-=z ^^ ; then -ji^=-!^±^, and the formula Z becomes F 
 L-::ii^=L(w + «)— L»? = 2a(^V + 7^T:rT3 + -7-^— + Scc) F 
 
 m \. ■ / I Vjm-J-« 3','''"+") J(»"'+"^' ' 
 
 c/0. Letrt=— ^; then '+"=1^!!-, and the formula Z becomes G 
 
 L(-^) = L?n—L (m—;i) = 2u,(-^-.+ — ^,t+ . "' ,, + 8cc) G 
 
 6/0. Let azz^" : thenl±:l = ^ and the formula Z becomes H 
 
 L^ = L;«-L;;=:2.4(^) + '(,:5^)'+,'.(^)'-f&c) - - H 
 
 'jmo. Let -^be fubftituted for - in the formula B : let this new formu- 
 
 / iti - til 
 
 la be divided by f; and Let h[m' — //') or L[m-{-f/)-{-L)in — nzza- and 
 L[m-{-!i)L.{m-= — ;/) s : then fliall 
 
 REMARKS. 
 
 7. Of three quantities 771 — //, ;/; and /«+"» ^^ arithmetical progreflion, 
 tlic logarithm of the fecond, being given the logarithms of the other 
 two may be found by one operation, if the odd and even powers of 
 -^in the fcricfcs A and R arc calculated apart. 
 
 E c 8.
 
 io8 APPENDIX. 
 
 8. If n is fuppofcd equal to unity, and if jt* (the modulus of the 
 fyftem of logarithms to be afterwards determined), confifts of a great 
 number of figures, it will be much more convenient, in calculating by 
 the feriefes A, B, C, D, F, and G, to confider ^. as the numerator of each 
 term than as the multiplier of the fum of the terms. 
 
 9. The firfl flep -^^"- of the feries F will give the logarithms of all 
 numbers greater than 20000 true to fifteen places, if thofe of all num- 
 bers lefs than 20000 are given, and if 2/x.« does not exceed a few units. 
 
 10. The firfl ftep t +-^ of the feries i will give the logarithms of 
 all numbers. greater than loooo true to nineteen places, if thofe of all 
 numbers lefs than loooo are given, and if?; does not exceed a few units. 
 
 The reader will eafily fee that the logarithm of all numbers below 
 VI being known, that of ^±^ and confequently that of m-\-n and there- 
 fore (T as well as ; will be known. 
 
 1 1. Various methods might be taken to compute with eafe the lo- 
 garithms of the lower prime numbers. The logarithms, for example, 
 of about two tliirds of the primes under 1 00 may be obtained with lit- 
 tle trouble from a table of the continual halfs of the modulus, n being 
 = I. The infpedion of the following table will make this evident. 
 
 given
 
 APPENDIX. 
 
 109 
 
 given Ifought 
 
 
 1 ^ 
 
 
 fcriet 
 
 
 1 
 
 ' W-f-I 
 
 m — I 
 
 the logar of. 
 
 I 
 
 2 
 
 2 
 
 
 I 
 
 B 
 
 2 
 
 3 
 
 2 
 
 3 
 
 
 A 
 
 2 and 3 
 
 5 
 
 2' 
 
 5 
 
 3 
 
 C 
 
 2 and 3 
 
 7 
 
 2' 
 
 3' 
 
 7 
 
 C 
 
 2,3 and 5 
 
 17 
 
 2* 
 
 >7 
 
 3X5 
 
 C 
 
 2 and 3 
 
 1 1 
 
 2' 
 
 3X11 
 
 
 A 
 
 2 
 
 3' 
 
 2' 
 
 
 31 
 
 B 
 
 2,3,5 and 7 
 
 '3 
 
 2« 
 
 5X13 
 
 3*X7 
 
 C 
 
 2 and 3 
 
 43 
 
 2' 
 
 3X43 
 
 
 A 
 
 2,3,5 and? 
 
 19 
 
 2 XIO 
 
 3x7 
 
 '9 
 
 C 
 
 2,3,5 »"d 13 
 
 4' 
 
 2' XIO 
 
 ^I 
 
 3X13 
 
 C 
 
 2,3 and 5 
 
 79 
 
 2» XIO 
 
 3* 
 
 79 
 
 C 
 
 2,5 and 7 
 
 23 
 
 2-'XlO 
 
 7X23 
 
 
 A 
 
 2,3 and 5 
 
 53 
 
 2*XlO 
 
 
 3x53 
 
 B 
 
 2,5 and 1 1 
 
 20 
 
 2'XIO 
 
 1 
 
 11 X29 
 
 B 
 
 2,3 and 5 
 
 71 
 
 2* XIO 
 
 
 3'X7i 
 
 B 
 
 2,3,5 and 7 
 
 61 
 
 2' XIO 
 
 3X7X61 
 
 
 A 
 
 12. The value of L(i + 2) was firfl given by Nicolas Mercator, who 
 deduced it from a property of the cqviilateral hyperbola*. The feries 
 c was firft d em on fl rated by James Gregory f. A Tories fome what lefs 
 general than / was produced by John Keill.in his treatife de Naiura and 
 arithmetica hgar'ithmoriwi : but I think I have fome where feen it attri- 
 buted to Newton. Some of the other formulce I believe are new. 
 
 PROBLEM II. 
 
 13. To exhibit the modulus of af\J}an of logarithm!. This is efTcdled by 
 
 fubftituting c for w, and i for //, in the equation H. Its value is as. 
 
 follows : 
 
 1 
 
 2(7^)+;(;{p{)' + '(:f') 4-^< 
 
 UEMAUKS, 
 
 • Logaritlimotechnia. f Exer. Gfom.
 
 fii APPENDIX. 
 
 REMARKS. 
 
 14. In our common fyflcm of logarithms, c is equal to 10; which gives 
 the following values of w, and its reciprocal to thirty decimal places. 
 
 /x=: 0.4342 9 44819 03251 83765 11289 ^^9^7 
 i =2.30258 50929 94045 68401 69914 54684 
 
 15. The modulus of Napier's fyflem is unity: for he fuppofed the 
 logarithm of a number diflering from unity by a very fmall quantity d 
 to be equal to the fum or difference of i and ^: Hence if ^L denote the 
 common, or Brigg's, logarithm, and 'L, Napier's logarithm of the fame 
 number; then 
 
 ^L = (0.43429 Sec) 'L; and 'L=r (2.30258 &c) ^L 
 
 PROBLIJVI. III. 
 
 1 6. To exhibit the number of a given logarithm. We have feen that d be- 
 ing — 1^ and jj. a finite quantity, that / —\+ -, (5) : we have 
 
 /i 
 
 therefore c^ — ^x +-)'', and confequently 
 
 and if x is negative, 
 
 f-'^=i— 1 + -A— ^^ + &c -T 
 
 Hence, by dividing O by "^j 
 
 2.V '+7-+Tl5-+-— ->« &C 
 
 fv " — — 
 
 o 
 
 17. If -v is greater than ^m. the above feriefes converge fo flowly that 
 that they are of no ufe for finding the number correfponding to a gi- 
 ven
 
 APPENDIX. Ill 
 
 given logarithm. Let dierefore m and n be two numbers di^cring little 
 from each other, in being greater than », and 
 
 I mo. Lctx-=L{^)=hm—Ln=i. Thenc'rr-^ and c—'=± and the 
 equations O and "Y give 
 
 m=r,{i-\-i + 4^^+Ti:-^+ Sec) - - - - M 
 
 "='"{'-^-\-T£r^-T:,B+ ^^) - - - - N 
 2do. Let .v = L (^)^=>L(^) = ;L;«-4L« = :; : then/' = ^ and the e- 
 quation CL gives • 
 
 I S. More generally, let there be any number « of numbers m'\-m'^ 
 
 '\m"'^m""s: ^fn'"' which, taken confecutively, differ little from each 
 
 other: and let Lw'^'— -L;«'^-^'=jl^^l and j2-(«— 2)/'^+lf=ili;=l),"> 
 
 I'— 1 1 
 ±i/'=: a'^' (the quantities 1/>|, ^^ — 1|, M, ?i — 1| &c. inclofcd in /inei, ex- 
 
 preffmg funply fome terms of the feries i, 2, 3, 4, 5 Sic) : we have 
 
 "" It: jV"'" ■ (— ^H"-.^W"" .^M A' A" , A'" , g^c o 
 
 m' ' 
 
 REMARKS. 
 
 19. If the logarithms of the firfl: 20000 natural mimbers are given, 
 the two firfl ftcps of the feries «(i+-^+_iL.) of the feries M, or «('~-J- 
 
 ■t-- V.v ) of the feries N, or the firft ftep « (17:^7) of the feries P, will give 
 the number m or ?; true to about the fourteenth decimal place. 
 
 r f 20.
 
 1 12 
 
 APPENDIX. 
 
 zc. The fciiefes M and N were firft given by Halley, in the Philofo- 
 phical tranfaclions for the year 1C95. He exiiibited alfo a fcries the 
 (lime •witli r, but under an inelegant form ; probably owing to his hav- 
 ing deduced it from the aclual diviiion of M by N, 
 
 PROBLEM IV. 
 
 21. T'o cxijibk the mimhcr ivbofe logarithm is equal to the modulus. Tliis 
 is effe(5led by tlae fubflitution of (i. for x in the formula ^^. It's value 
 is as follows 
 
 # 
 
 
 or taking the fum of thirty fradlional terms 
 
 ^'^ = 2.71828 182S4 59045 23536 02S74 71353 
 
 II.
 
 IL 
 
 A TABLE OF NTAPIER's LOGARITHMS, 
 
 ©F ALL THE NATURAL NUMBERS FROM I tO lOI tO TWENTY 
 
 SEVEN PLACES. 
 
 Num. 
 
 I it 
 
 17 
 
 Logarithms. 
 
 o.ccooo.ocooo.coooo.oocoo.ocooo.o 
 0.693 14. 71 805. 5994.5. 3C94I.72:j2 1.'. 
 
 3 1 .09861 .22886. 68109. 69 139. 5245 2. 4 
 
 4 1 .38629,4361 1. 1 9:^90. 6 1 883. 44642. 4 
 
 5 1. 60943. 79124. 34100.37460. 07593. 3 
 
 ' •79'75-94'^'92' 28055.00081 .24773.6 
 1.94591.01490.55313.30510.53527.4 
 
 8 2.07944.154.6. 79335.92^2^.16963.6 
 
 9 2. 19722.45773. 362 1 9. 382 79. 049C4.S 
 2. 3C258. 50929. 94045. 68401.79914. 6 
 
 II 2.3v739.5272:-9S370-y44o<5-i9435'-S 
 1 2 2.4!i'490.'''6497. 1'Sooo. 31022. 97054. 8 
 
 •3 2-5<^494-93574-'S'53^-7.^6o5..hS74.4 
 
 14 2.6 .^^,-.73:96, 15258.61452.25848.0 
 
 15 2.7c' : c. 020 II. 02 2 10. 0^1599. 60045. 7 
 
 2.7 7: 58.8; 2 22. 397 Si. 237 ',6. 89 2^4. () 
 2.^332l .33440. 56216.08024.95346.2 
 i.f'9'^37'' '57^' 961C.4. 6922^.77226.0 
 2.94443.89791.66440.46000.90274.3 
 2.90,57^.2^735. f3y90-''&343-S = ^.'5-^ 
 
 Num. 
 
 Lo 
 
 G.VRITHMS. 
 
 21 3.C44J^-2+.n7- :34"-S9650-0S979.8 
 
 22 3. 09 104. 24533. 583 15. 85347. r; 1757.0 
 
 23 3- '354v 42 '59- *9' 49- 69^80. 67528. 3 
 
 24 3. 1 7805. 3S3C3. 47945. 61964. 69416,0 
 
 25 3.21887.58248. '18200.74920. 15 186. 7 
 
 26 
 
 27 
 28 
 29 
 
 3' 
 
 32 
 33 
 3 + 
 35 
 
 3^ 
 
 37 
 
 39 
 
 ao 
 
 3.25809.6:380. 2 1482.04547.01-95.6 
 3.29583.68660. 04329. 074 I?. 57357. I 
 
 3.33220.45101. 75203. 92393. 981 69. S 
 
 3. 36729. 58299. 86474. 02718.3272--. 3 
 3.40119.73816. 62 155. 37541. 32366. J 
 
 3.43398.72044.85146 .24592.91643.3 
 
 3. 4''^573-59°27. 99726.54708.61606. i 
 3. 49650. 756 14. 664<o. 2 3 545.71^.88.1 
 3.52636.05246. 16161. 38^66.67667. + 
 3. 555 54- ^-•614.894' 3 •57y7C'. 61 1 20. 8 
 
 3-58351.89384.56110.0016:. 49547. 2 
 3. 6109 1. 79 1 26. 4422 <. 4443'". !<- 956. 7 
 3. 63758. 6i5<;7. 25385. 76^42. '125^5. J 
 }.()(>}^6. 16461 . 29646.42744.87326.8 
 3. 68SS;. 94541. 139^6. 3028 J. 24557.3 
 
 ^»
 
 114 
 
 APPENDIX. 
 
 Num. 
 
 Logarithms. 
 
 4« 3. 7'35"-'(^<567. 04507. 80386. 67633. 7 
 
 4? 3. 73 766. 96 1 82. 8 ^368. 3059 1. 7S30 1.0 
 
 4 J 3. 76 120. or 156.93562.42347.28425.2 
 
 44 3.784iS,> 6339. 18261 . 16289.64078,2 
 
 45 3.80666.24897.70319.75739.12498. I 
 
 ^6 3,82864. 13964. ^909;. 00022. 39849. 5 
 
 47 3.^SO'4-76®'7- 10058.58682.09506.7 
 
 48 3.87120. 101C9.07H90. 92906,^1 73 7. 2 
 
 49 3.89182.02981. 10626.6102 1 .07054. 8 
 
 50 3.91202,30054, 28146.05861.87507.9 
 
 51 3.93182.56327.24325,77164.47798.6 
 
 52 3 -95 '24-37 '85. 8 1427 -35+88. 7 95 '6. 9 
 
 53 3.97029.19135.52121.83414.44691.4 
 
 54 3.988^8.40465. 64274.38360.29678.3 
 
 55 4.00733.31852.32470.91866.27029. t 
 
 56 
 
 57 
 58 
 59 
 60 
 
 61 
 (>i 
 63 
 64 
 65 
 
 Num. 
 
 Logarithms. 
 
 4.02535.16907. 35M9-23335-7349'- ' 
 4.04305.12678. 34550. 15140.42726.7 
 4. 06c 44. 30 105. 46419.33660.05041.6 
 
 4-07753-74439- 057 ' 9- 45of>' •60503-8 
 4.09434.456:2. 221C0.684S3.046S8. 1 
 
 4.11087.3864T. 7331 1.248-5. 1 3891. 1 
 4. 127 13. 4^85o. 45091.55 534. 63964.5 
 
 4.1431^47263.91532.68789.58432.2 
 
 4. 15888.30833. 5c,:67 1,85650. 33927. 3 
 4. 1 7438. 7 2698. 95 637. 1 1065. 42467. 8 
 
 66 4.18965.47420. 26425.54487.44209.4 
 
 67 4. 20469.26193, 90966.05967.00720.0 
 63 4. 2 1 950. 7 705 1 . 76106.69908. ^9988. 6 
 
 69 4.23410.65045. 97259.38220.19980.7 
 
 70 4.34849.5 242c. 49358.98912.33442.0 
 
 71 I 4.26267.98770,41315.42132.94545.3 
 
 72 
 
 73 
 74 
 75 
 
 ^6 
 77 
 78 
 
 79 
 
 80 
 
 86 
 
 87 
 88 
 
 8y 
 90 
 
 9' 
 92 
 
 93 
 94 
 95 
 
 96 
 
 97 
 98 
 
 99 
 100 
 
 4.27666.61 190, 1(^055.31 1 04. 2 1 868.4 
 4.29045.9441 1 . 48391 . p 2909.2 1088.6 
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 101 I 4.61512.05168.41259.45088.41982.7
 
 III. 
 
 TRIGONOMETRICAL THEOREMS. 
 
 (i) Lemma i. The produ<fl of the radius by the difFerence of the 
 verfed fines of two arcs is equal to twice the product of the fines of 
 half the fum and half the difFerence of thofe arcs. 
 R (fin V, a— Cm. Y,b) = 2 findi^x fin^. 
 
 (2) Corollary. The produd: of the radius by the verfed fine of an 
 arc is equal to twice the fquare of the fine of half that arc. 
 
 R fin V, rt 1= 2 fin ' l-a. 
 
 (3) Lem. 2. The fum of the cofines of two arcs is to their difference 
 as the cotangent of half the fum of thofe arcs is to the tangent of half 
 their difference. 
 
 Cof a-|-cof3:cof^ — zo^ b: ; cot^ : tangtr?. 
 
 (4) Lem. 3. The fum of the fines of two arcs is to their difference as 
 the tangent of half the fum of thofe arcs is to the tangent of half their 
 their difference. 
 
 Sin a -f- fin l> : fin a — fin b : : tang i±.* : tang ^. 
 
 (5) Lem. 4. The fum of the cotangents of two arcs is to tlicir difTe- 
 rence as tlie fine of the fum" of thofe arcs is to the fine of tliclr difference 
 
 Cot a -\- cozb: cot a— cot b : : {in{b-{-j) ; fin {6 — a), 
 
 G g (6)
 
 ti6 APPENDIX. 
 
 (6) Lem. 5. The produdl of the fine of the fum of two arcs and the 
 tangent of half that fum, is to the produdl of the fine of their difference 
 and the tangent of half tliat difference, as the fquare of the fine of half 
 their fum is to the fquare of the fine of half their difference. 
 
 Sin {a+b) X tang ^ : fin {a—b) x tang ii^ : : fin' '4^ : fin' ^*. 
 
 (7) "Lem. 6. The produifl of the fine of the fum of two arcs and the 
 tangent of half their difference, is to the producft of the fine of their dif- 
 ference and the tangent of half their fum, as the fquare of the cofme of 
 half their fum is to the fquare of the cofine of half tlieir difference. 
 
 Sin {a-]-b) X tang i"^ : fin {a—b) x tang "4^ -. : cof ' "4^ : cof ' "7*. 
 
 (8) Lem. 7. In right angled fpherical triangles the cofine of the hy- 
 pothenufe is to the cotangent of one of the oblique angles as the cotan- 
 gent of the other is to the radius. 
 
 (9) Lem. 8. In right angled fpherical triangles the cofine of the hy- 
 pothenufe is to the cofine of one of the fides as the cofine of the other 
 is to the radius. 
 
 (10) Lem. 9. In any fpherical triangle the produdl of the fines of 
 the two fides is to the fquare of the radius as the difference of the ver- 
 fed fines of the bafe and the difference of the two fides is to the verfed 
 fine of the vertical angle, Fig. XiV. 
 
 Sin ABx fin BC : R': : fin V, AC— fin V, (AB— BC) : fin V, B * 
 
 (11) Lem. 10. In any fpherical triangle the produ(ft of the fines of 
 the two fides is to the fquare of the radius, as the difference of the ver- 
 fed fines of tlie fum of the two fides and the bafe is to the verfed fine 
 of the fupplement of the vertical angle. Fig. XIV. 
 
 Sin AB X fin BC : R': : fin V, (AB + BC)— fin V, AC : fin V, fup. B. 
 
 * Tliis is one of Regiomontanus' propofitionj.
 
 APPENDIX. ri7 
 
 (12) The natural parts of a triangle are its three fides and its three 
 angles. 
 
 ( 1 3) The circular parts of a redangular (or quadrantal) fpherical tri- 
 angle are the two namral parts adjoining to the right angle (or qua- 
 drant fide) and the complements of the other tlu^ee. 
 
 (14) Any one of thefe five being confidered as a middle part, the 
 two next to it are called the adjacent parts, and the other two the op- 
 pofite parts : Thus, in the triangle <^AB (Ing. XV.) rectangular in A, 
 if tlw; complement of the angle d is taken as a middle part, the adjacent 
 parts are the fide d A and tlie complement of the hypothenufc db ; and 
 the oppofite parts the fide, b A and the complement of the angle /-. 
 
 (15) Of five great circles of the fpherc AB, BC, CD, DE, and EA 
 (Fig. XV.) let the firfi: interfecl the fecond ; the fecond, the third ; the 
 third, the fourth ; the fourth, the fifth; and the fifth, tlie fii-ft; at right 
 angles in the points B, C, D, E and A : there are formed, by tlie inter- 
 fecflions mentioned and by thofe at the refpcclive poles a, b^ c, d and e 
 of thefe great circles, five rectangular triangles dXb^ bDe, eBc, cEa and 
 aCd : and, if thefe poles are joined by the qviadrantal arcs ab^ be, cd, de 
 and ea, there are formed five quadrantal triangles adb, dbe, bee, ecu, and 
 cad. The circular parts in all thefe triangles are the fame : tlie pofttion 
 of thefe equal circular parts with refpe(fl to one another in each of thefe 
 triangles is different : therefore 
 
 (16) What is true of the circular parts of a redangtdar triangle is 
 true of thofe of a quadrantal ; and what is true of one middle part and 
 its adjacent and oppofite parts is true of the other four middle parts and 
 their adjacent and oppofite parts. 
 
 (^7)
 
 ii8 APPENDIX. 
 
 ( 1 7) The circular parts of an oblique fpherical triangle are its tlirec 
 fides and xht fupplcmcnts of its three angles. 
 
 (i8) Any one of thcfe fix being confidered as a middle part, the two 
 next to it may be called the adjacent parts ; the one facing it, the re- 
 mote part ; and the other two, the oppofite parts ; Thus, in the triangle 
 ABC (Fig. XIV.), if the fide AC is taken as a middle part, the adjacent 
 parts are the fupplcments of the angles A and C ; the oppofite parts, 
 the fides AB and BC, and the remote part, the fupplement of the angle B. 
 
 ( 1 9) Of fix great circles of the fphere let the firft three, AB, BC, and 
 CA, interfedl each other at the poles, B, C and A, of the fecond three, 
 ca^ ab and be : the interfedlions, <:, a and b^ of the latter are the poles of 
 the former : there are formed two triangles ABC and abc in which tlie 
 circular parts are the fame ; the pofition of thefe equal circular parts is 
 different in both : therefore 
 
 (20) What is true of one middle part and its adjacent, oppofite, and 
 remote parts, is true of any other middle part and its adjacent, oppofite, 
 and remote parts. 
 
 (21) If an arc h^Yid pafs through the vertices of thefe two triangles, 
 it will be perpendicular to their bafes CDA and cda^ and the fegments 
 at the bafc of the owz triangle will be the complements of the fegments 
 at the vertical angle of the other: that is, CD = 90° — dha^ AD = 90° — 
 dbc, f^=9o°— ABD, ^^/=9o°— DBC. 
 
 (22) If the radius of the fphere is fuppofcd infinite, the fines and tan- 
 gents of the fides of a triangle defcribed on its furface, become the fides 
 themfelves of a plane triangle. Conf cquently all the formulae of fpheri- 
 cal trigonometry, where the fines and tangents only of the fides enter, 
 are applicable to plane trigonometry. Thofe, however, in which any 
 
 fundions
 
 .APPENDIX. 119 
 
 fundllons of all the three angles and only one fine or tangent of ous 
 fide enter, mufl be excepted. 
 
 (23) Of the circular parts we fhall denote the middle one by M, the 
 adjacent ones by A and ^, and the oppofite ones by O and 0. If the tri- 
 angle is oblique, the remote part we fliall call m, and the fcgments at 
 a fide or angle (21)5 and s. 
 
 (24) Theorem i. Of the circular parts (13) of a rectangular (or quad- 
 rantal) fpherical triangle, the produdl of the radius and the fine of the 
 middle part, the produ<5l of the tangents of the adjacent parts and the 
 product of the cofines of the oppofite parts, are equal. 
 
 Demonflration. In the right angled fpherical triangle d\b (Fig. XV.) 
 we have cof bd : cot. Abd : : cot Adb : R (8), and cof bd : cof Ai : : coCAd: 
 R (9) ; therefore R X cof bd=cotAbdx cot Adb = cofAbxcorAd; there- 
 fore (16) 
 
 R X fin M = tang A x tang a z= cof O X cof 0. 
 
 {1^) Corollary i. In any fpherical triangle, the fines of the fides arc 
 proportional to the fines of the oppofite angles. For, in the right angled 
 triangles ADB and CDB (Fig. XIII.), Rxfm BD = fm ABxfm A, and 
 R X fin BD =fin EC X fin C ; therefore fin AB : fin BG : : fin C : fm A 
 
 (26) Cor. 2. In any fpherical triangle, the fines of the fegments of 
 one of its fides (produced if neceflary) are proportional to the cotangents 
 of the angles at the extremities of that fide. For, in the right angled 
 triangles ADB and CDB, Rx fin AD = cot Ax tang BD and R x fin DC 
 = cot Cx tang BD y therefore fin AD : fm DC : ; cot A : cot G 
 
 (27) Cor. 3. In any fpherical triangle, the cofines of any two fides 
 are proportional to the cofines of the fegments of the third fide. For, 
 in the right angled triangles ADB and CDB, Rx cof AB=:cof ADxcof 
 
 H h • DB,
 
 120 APPENDIX. 
 
 DB, and Rx cof BC=:cof CDx cof DB ; therefore cof AB : cof BC : : 
 cof AD: cof DC 
 
 (28) Remark i. This theorem ferves for the folutionof all pofliblc 
 cafes of reclanglar or quadrantal fpherical triangles, and for the folutiou 
 of all poffible cafes of oblique fpherical triangles (by means of the arc 
 drawn from one of its angles perpendicular on the oppofite fide) ; ex- 
 cepting when the three angles, or the three iides only, are the data. 
 
 (29) Rem. 2. This theorem,, by confining the middle part to the two 
 fides, (22) ferves alfo for the folution of all pofliblc cafes of redlangular 
 plane triangles, and for the folution of all poflible cafes of oblique angled 
 plane triangles (by means of the perpendicular drawn from an angle to 
 to the oppofite fide) ; excepting when the three fides only are the data. 
 
 (30) Rem. 3. Were the complements of the two parts adjoining to 
 the right angle or quadrant fide and the other three natural parts taken 
 as the circular parts, the theorem would be, 
 
 R X cof M =z cot A X cot a — fin O X fm 0. 
 But the other is preferable, becaufe it is more eafily remembered. The 
 fecond letter of the word tangent is the fame with the firft of adjacent. 
 It is the fame of the words cofine and oppofite. If this is attended to, 
 it is hardly poffible to forget the enunciation of the theorem. 
 
 (31) Theorem 2. Of the circular parts (17) of an oblique fpherical 
 triangle, the fquare of the fine of half the middle part, is to the fquare 
 of the radius ; as the producl of the fines of half die fum and half the 
 difference of the fum of the adjacent parts and the remote part, is to the 
 produ<ft of the fines of the adjacent parts. 
 
 Dem. For fince (Fig. XIV.) fin V. fupp. B : R': : fin V, (AB+BC)— fin 
 V, AC;fm ABXfinBC (11), it follows that fm'^ fupp. B: R':: fin 
 
 t .xB-\~liC JfAr. \
 
 APPENDIX. ,ci 
 
 ■ (JF+Bc+Ac) xfin {dl±£l-l£) fmABxfmBC(2and i) ; therefore (20) 
 Sin*^- M : R\ : fin (^Ed+Il) x ^"1" ('iii^::!') : fin. Ax fin a. 
 
 (32) Theorem 3. Of the circular parts of an oblique fpherical tri- 
 angle, The fquare of the cofuie of half the middle part is to the fquarc 
 of the radius ; as the produ(5l of the fines of half the fum and half the 
 difference of the remote part and the difference of the adjacent parts, is 
 to the produdl of tlie fines of the adjacent parts. 
 
 Dem. For fince fin VB : R' : : fin V, AC— fin V, (AB— BC) : fin AB 
 +rin BG (10), it follows that cof "i fupp. B : R': : {'m^±^J=-J£X fin 
 
 (d£=d^EiE.) : ABXfinBC (2) and (i) ; therefore (20) 
 
 CoCi M : R' : : fin('"+^="^) X fin { '--^ : fin A X fin n. 
 
 {22) Theorem 4. Of the circular parts of an oblique fpherical tri- 
 angle, The fquare of the tangent of half the middle part is to the 
 fquare of the radius ; as the produdl of the fines of half the fum and 
 half the difference of the fum of tlie adjacent parts and the remote part, 
 is to the produdl of half the fum and half the difference of the remote 
 part and the difference of the adjacent parts. 
 
 That is (by comparing the two preceding theorems) 
 
 Tang'^ M : R' : : fin p+^+" ') x fin(^:::^) : fin['2±dEi) X fin('.2r-.-i=f) 
 
 (34) Theorem 5. Of the circular parrs of an oblique fpherical tri- 
 angle. The produ<5l of the tangents of half the fum and half the diffe- 
 rence of tlie fegments of the middle part is equal to the produdl of the 
 "•tangents of half the fum and half the difference of tlie oppofite parts. 
 Dem. For fince cof BA: cof BC :: cof DA: cof DC (27) it follows 
 that cof BA+cof BC: cof BA— cof BC : : cof DA+cof DC: cof DA— 
 
 cof
 
 122 APPENDIX, 
 
 cof DC J therefore (3) cot (^-^i) : tang {S^) : :cot(<i^^) :tan-(i^') ; 
 therefore tang (^I£±iL^) x tang (^^0 = = =tang (££±±?)Xtang {^^) ; 
 tlicrefore (20 and 21) 
 
 Tang (^+0X tang (^r)=tang (°±f)Xtang {2^) 
 
 {^$) Rem. 4. By any of the theorems 2, 3, or 4, being given the 
 three fides or three angles of a fpherical triangle, may be found any of 
 its angles or fides ; and, confining the middle part to the fupplement 
 of an angle, being given the three fides of a plane triangle, may be 
 found (22) 
 
 (36) Rem. 5. By theorem 5, being given the tliree fides or three angles 
 of a fpherical triangle, the fegment of any of its fides or angles may be 
 found ; and confining the middle part to a fide, being given the three 
 fides of a plane triangle, the fegments of any of its fides may be found. 
 
 (37) Rem. 6. By the firft theorem, and any one of the other four, 
 may be folved all the poffible cafes of fpherical and plane triangles. 
 Of thefe four, the lafl is the molt elegant and the moft eafily re- 
 membered. 
 
 (38) Theorem 6. Of the circular parts of an oblique fpherical tri- 
 angle, the tangents of half the fum and half the difference of the feg- 
 ments of the middle part are proportional to the fines of the fum and 
 the difference of the adjacent parts. 
 
 Dem. For fince fin CD : fin DA : : cot C : cot A (26), it follows that 
 fin CD + fin DA : fin CD — fin DA : : cot C-j-cot A : cot C — cot A ; there- 
 fore tang (^^) : tang {'-^) : : fin (A+C) : fin (A— C) ; therefore 
 
 (20 and 21) 
 
 Tan {'-±1) : tang (V) = •• fm (A+^) : fin (A~^) 
 
 {39)
 
 APPENDIX. 12^ 
 
 (39) Rem. 7. By diis theorem, being given two fides and the in- 
 ckided angle, or two angles and the included fide of any triangle, the 
 fegm.ents of the angle or fide may be found. 
 
 (40) Theorem 7. Of the circular parts of an oblique fpherical triangle, 
 The tangents of half the flim and half the difference of the adjacent parts 
 are proportional to the tangents of half the fum and half the difference, 
 of the oppoiite parts. 
 
 Dem. For fmce lin BC : fm BA : : fm A : fin C (25), it follows that 
 fm BC+ fm BA : fm BC— fm BA : : fm A+ fm C : lin A— fm C, there- 
 fore (4) tang (^£±^) : tang (££=£i') : : tang {d±S) : tang (--=£) : there- 
 
 fore (20) 
 
 Tang {d±l) : tang (-1=1) : : tang {2±l) : tang (2=L), 
 
 (41) Rem. 8. By this theorem, being given two fides and the includ- 
 ed angle of a plane triangle (22), the other angles may be found. 
 
 (42) Theorem 8. Of the circular parts of any fphericaljiriangle. The 
 tangents of half the middle part and half the difference of the oppofite 
 parts are proportional to the fines of half the fum and half the dif- 
 ference of the adjacent parts. 
 
 Dem. For fince tang {'-±1) X tang ±=f)rr tang ('?±?) x tang {9^%{;>^). 
 
 and tang (:y:i) : tang (^) : : fm (A+«) : fm {A—.i), (38); and tang 
 
 2±1 : tang ^^-=2 : : tang (dp) -. tang (i^),(4o) it follows, that tang' (H:'). 
 
 tang' (^) :: fm {A-{-a) x tang ±tl : fin.(A— ^)x tang (■'=:); 
 
 therefore (6) 
 ' _ Tang i M ; tang (2=?) : ; fin (iif) : fin (:i=;). 
 
 (+3) 
 
 I 1
 
 124 APPENDIX. 
 
 (4'^) Theorem 9. Of the circular parts of an oblique fpherical triangle, 
 The tangents of half the middle part and half the fum of the oppofite 
 parts are proportional to the cofmcs of half the fum and half the dif- 
 ference of the adjacent parts. 
 
 Dem. For fmce tang (^) X tang {'^) = tang (2±f ) x tang [2=2], (34) ; 
 
 and tang (^) : tang (^) :: fm (A-f^) : fm (A — a), (38); and tang 
 9=1 : tang 9±1 -, -. tang ±^ : tang ^' (40) ; it follows, that tang * 1+/- 
 tang* ^: :fin (A+«) tang ^^:fm (A— ^) tang (^) : therefore (7) 
 Tang iM : tang 2±1 -.-.cof^tl :.cof±^. 
 
 (44) Rem. 9. From thefe two theorems it is evident, that, being gi- 
 ven two angles and the inclvided fide, or two fides and the included 
 angles of any fpherical triangle, the other two fides, or the other two 
 angles may be found ; and being given two angles and the included fide 
 of any plane triangle, the other two fides may be found by iwo analogies 
 only. 
 
 From tliefe proportions are deduced the following 
 
 TRIGONOMETRICAL FO RMULiE. 
 
 (45) In any fpherical triangle ABC, Fig. XIV. we have 
 
 Sin AB X fin BC : R': : fin ^c+/ib—bc x fin ac—ab—hc -. fm' jB (32) 
 
 Sin AB X fin BC : R* :: fin yiB+jiC+ACxfin ab+bc—ac -. cof '^6,(3 1 ) 
 
 Sin ab+bc+jc X fm yJ/i+nc—/!C : R' : ; fm ac+ab—bc x fin 
 
 jc—AB—Bc : tang' I B {2)i) 
 
 '■*' 
 
 Sin Ax fin C : R"' : : — cof ^+ c+ ^ x cof ^1+ c— B : fin ' ^ AC 
 
 Sin Ax fin C ; R'" : : cof £+£^x cof ^-^ ; cof I AC 
 
 a a 
 
 coi
 
 APPENDIX. i.. 
 
 Cof£+^X cof£-^^ : R' : : — cof £+£+5 x cof w+£_£ ; tangVf AC 
 Tang .;, AC : tang ^C+7iA . . ^^ng b c—ba . j-^ng e n— da 
 
 * * a 
 
 Sin (A+C) : fin (A— C) : : tang I AC : tang en— da 
 
 i 
 
 Cot i B : tang ^+^ : : tang jtz£j tang cdb-dba 
 Sin (BC+BA) : fin (BC— BA) : : cot ^ B : tang c.bd-dba 
 Tang J^c+BA , tai^g ^g-^ // ; ; tang ^+c : tang ^-^ 
 Sin ^+c . fin ^-C ; : tang I AC : tang ^C-ba 
 €of ^+^ : cof ^-C : : tang \ AC : tang ^g+^-^ 
 
 Z Z 1 
 
 Sin I£±^A: fin -^^-^-^ : : cot I B : tang ^-^ 
 Cof BC+B/i . cof ^c-z?/r . : cot i B : tang ^+c 
 
 2 1 2 
 
 (46) In any plane triangle ABC, Fig. XVI. we have (22) 
 
 AB X BC : R' : : ( ac+ab-Bc ) X { AC-^n—Bc ) ; {n\ \ B 
 
 2 i 
 
 AB X BC : R' : : { AoTJuJ+Ac ) x {ab+bc—ac) : coP 4 B 
 
 2 • 1 
 
 (7?A+5C+.YC) X [ab+Tc—AC) : R* : : (/^C+^^^^^t;) x [AC—AB^^FC) 
 
 : tang' | B 
 AC : BC+BA : : BC— BA : CD— Dx\ 
 Sin (A+C) : fin (A— C) : : AC : CD— DA 
 BC+BA : BC— BA : : tang d±L : tang .^t±. : : cot ^ B : tang jtS. 
 
 : : cot ^- B : tang CDB-DBA 
 
 2 
 
 Sin ^+C , fin ^~C : : AC : BC— BA 
 
 2 2 
 
 C^f j1±£. : coi±z£. : : AC : BC+BA . 
 
 » a 
 
 rv.
 
 J \ l
 
 tktuama 
 
 IV. 
 
 THE HYPERBOLA AS CONNECTED WITH THE LOGARITHMS. 
 
 1. While a rtralght line PM (Fig. XVII.) moves parallel to itfelf 
 along the indefinite ftraight line CPD with a velocity always proportional 
 to the diflance of its extremity P from a fixed point C, let its other 
 extremity M approach to or recede from P, lb that PM may defcribc 
 equal fpaces in equal times : The point P will defcribe a part PP' or ?p' 
 of the Itraight line CD, while the point M defcribes a correfponding 
 part MM' or Mm of the curve ffz'SM'. 
 
 2. If the motion is luppofed to have begun at P, the area PM M'P' 
 or PM t/i'p' is the logaritlim of the abfcifs CP' or Cp\ 
 
 3. In order that equal fpaces may be defcribed in equal times, it is 
 evident that the greater or fmaller the abfcifs CP' or Cp' becomes with 
 regard to CP, the fmaller or greater mud the ordinate P'M' or p'm' be- 
 come with regard to PM ; Therefore CP' : CP : : PM : P'M', or Cp' : CP 
 : : PM :p'm' ; Therefore the producfl of any abfcifs by the corrcfpondent 
 ordinate is a conflant quantity : Therefore 
 
 4. The curve m'SM' is a hyperbola having CD for one of its affymp- 
 totcs, and C;, parallel to the ordinatcs, for the other. 
 
 Kk S'
 
 I ZO 
 
 APPENDIX. 
 
 5. From tills manner of conceiving the generation of the hyperbola 
 might be deduced the properties of that cm-ve and of the logarithms. 
 That CD and Ct, for inflancc, touch tlic curve at an inilnlte dlilance 
 from C appears from this : When the abfclfs is infinite, the ordinate 
 miift be zero, and vv'heii the abfclfs is zero, the ordinate mull be infi- 
 nite, in order tiiat their producl may equal the finite quantity PM 
 X CP : And that the logarithm of CP is zero appears from this ; PM is 
 length vfithout breadth and therefore no fpace. 
 
 6. Let CP = «, PM=//., PP'=.v and VM'zzy; we have (3);' = —, or, 
 developing the fraclion -' - in the manner firft taught by Nicolas Mer- 
 cator-'*, 
 
 ' a a^ a-' 
 
 7. It is evident that the fpace PMMT' is equal to the fum of all the 
 ordinates ■/ -\-y" -\-y" -{- ^c. on the abfclfs x. If the abfclfs is fup- 
 pofed to be divided into an iniinltc number of infinitely fmall and equal 
 parts, the abfciflk correfponding to the ordinates j'',j",j'"', Sec. may be 
 called I, 2, 3, Sec: therefore (6) 
 
 3/"z.,..(i-^+^-2^+&c) 
 therefore 
 
 
 •-■' i' Rcc 11 
 
 -7-"~""' ■^■— C<V • • • f 
 
 T3 
 
 ^ 
 
 4- &:c, 8cc. 
 
 * Logarithmotechnia 
 
 Now,
 
 A P P E N D I X. 1:9 
 
 Now, as was firft demoiiflrated by Wallis *, the fum i"-}- 2" -1-3" 4- Sec. 
 continued to infinity, that is to x' in this cafe, being equal to ■^' . ^^ 
 
 have 
 
 P]VlM'P' = L(^+x)=/.(x— il +3^i— 8cc) 
 
 and, if X is negative, 
 
 ?Mm'/ = L{a—x)=—^.{x + -(-^jL+ See) 
 
 8. The quantity y. depends on the angle DCj ^Oformed by tlie af- 
 fymtotes and the diftance MNzrw of the point M of the curve from, 
 the aflVniptote CD : as is evident from its value «, = ^. , , where r dc- 
 notes the radius of the circle. 
 
 • Aritli. Iiifihit. 
 
 SECTION.
 
 V. 
 
 PROPERTIES OF THE LOGARITHMIC. 
 
 1. While two points a and B are in S, (Fig. XVIII.) moving in 
 oppofite direclions along the indefinite (traight line CSD with a velocity 
 always proportional to their diflance from a fixed point C, let all the 
 points in SD and all the points in SC move in oppofite diredlions per- 
 pendicularly to CSD with any uniform velocity ; and in the inflant that 
 A or B paiTes through any point P' or p' let the point which left P' or 
 p' (lop in M' or in ; A and B will defcribe the a.xis, while the points that 
 move perpendicular to it, defcribe all the ordinates, or the area of the 
 curve wi'SM'. 
 
 2. This curve is called the logarithmic, becaufe its ordinate PM, P' 
 M' S-c. are the logarithms of its abfcifi[;E CP, CP', &c. 
 
 3. The ordinate C ■ , at the finite extremity C of the axis, is an afi^-mp- 
 tote to the curve : for, as the point that moves from S towards C can- 
 not arrive at C in any'finite time, the point that left C will move on 
 for ever. 
 
 4. The ordinate PM, a tangent to the curve at whofe extremity M 
 meets the point C, is called the modulus of the logaritlimic. We iliall 
 call PM the logarithmic modulus and CS the numeric modulus. 
 
 LI £,
 
 i:z APPENDIX 
 
 5. Let the portions Pr and PV of the axis be fuppofed defci'ibed in 
 equal times ; and let the ftraight lines Mf, M/ be drawn perpendicular 
 \o the ordinates cy., cr'^/ : we have CP : CP' :: P»: P'»' and vfji,=.i>'f^/ : 
 33 ut, if the equal tlnies are infinitely fmall, the arcs M^m, and M'/y/ are 
 ftraight linesand the right angled triangles CPM and Mf^w., fimilar ; con- 
 feqnently CP : PjNI : : M^ or Pr : v^jthercfore CF : PM : : P'x' : vu. or //. 
 
 6. To ^r^7Zf a tangent to any point AI' of the Logarithmic. Upon the or- 
 dinate P'M' take P'L'r=PM ; join the points C and L' and draw parral- 
 Icl to CL' the ftraight line MV meeting the axis in the point r' ; t'M' 
 touches the curve in the point M' : For fince (5) CP' : PM or P'L' : : P',' 
 or M'/ : v'm,', the triangles CP'L' and M'/^«.' are fimilar j therefore M'^/ 
 is parrallel toCL' ; therefore &.c : Hence, 
 
 7. The ordinates to the afTymptote, MQ^ MCV, &c. have for their lo- 
 garithms its abfcilFae CQ^ CQ^: and 
 
 8. The fubtangents CQ^ C'Q^, 8cc. upon the alTymptote are all equal 
 to the logarithmic modulus PM. 
 
 9. The fubtangent T'P' upon the axis is to the ordinate P'M' as the 
 abfcifs CP' is to the modulus PxM ; For the triangles T'P'M' and C'P'M' 
 are fimilar : Hence, 
 
 I o. The fubtangents upon the axis are to each other as the produ(fls 
 of the abfcifls and ordinates. 
 
 1 1. The fubnormal P'N' upon the axis is to the ordinate as the loga- 
 rithmic modulus to the abfcifs : For the triangles CP'L' and M'P'N' are 
 fimilar: Hence, 
 
 1 2. The fubnormals xipon the axis are to each other as the quotients 
 of the ordinates and abfciflx. 
 
 »3
 
 APPENDIX. 133 
 
 1 3 The fabtangent is to the fubnormal as the fquare of the abfcifs 
 to the fqirarc of jthe logarithmic modulus. 
 
 14. Let V\l = i/., CP =:s and P'M'=:;': and let z', z\z\ . . . s" be any 
 number of abfcifHr in geometrical progreflion ; S',S^\S"\. . . S ", the cor- 
 
 refpondent fubtangent-s, and <r', (r", <r"' o-", the correfpondciit fub- 
 
 normals upon the axis: -we have, (10) and (12) 
 
 «(«-!- i)(2//+i)/=:6(SV-fSV-fS"V"+....+SV'') 
 
 2;(i±5:)Cpj)=s;+s: s;:^ ^s- 
 
 1^^ \ I -\--Z J \ I 2 J (T a- or a- 
 
 ,„_sv s;^v' S'" X sv 
 
 ("illX -^ s" s'" . s" 
 
 I I J — , X "7/ X —TT, X X „ 
 
 \ jM' / C (T (T * (T 
 
 I ^. Let tlie numeric modulus CS-=m and SP'=:.v, the denominations 
 ofPMand P'M' continuing as before : we have P'' zzx and vX = v. 
 Now m+x : [/. :: X : y, (5) ; therefore _)- = ;;^ ; or if w= I y = ^>.x{^^ — 
 f/.x (i — x-f-.v* — x'-j-&:c) and therefore 
 
 jK = ^.(x — :- + li— &cc)r=Log. (i+x). 
 
 16. Let the area of any portion S'M'P' of tlie curve=A: wc have 
 
 • • • • • • 
 
 A = sv; therefore A =y ay zrs)'—3^'s: but _y = '^' (5)>or (15); therefore 
 
 Jj'z=/i/.z=:iy.~; therefore A = ~v — m.= -|-C: but when Azzo, then z=m 
 and ^ = 6; therefore ozz — f^m-{^C ; therefore Czzfjum and A = = {y — h)-|^ 
 /AW, that is 
 
 1 7. The area of any portion of the logarithmic is equal to the rec- 
 tangle under the abfcifs and the difference of the ordinate and the lo- 
 
 garithmiy
 
 »34 
 
 APPEND X. 
 
 garithmic modulus, togctlier with the rectangle under the moduli ; 
 Hence 
 
 1 8. The reclangle CL, under the moduli, is equal to tlie area SMP 
 contained by the logarithmic modulus PM, the portion of the axis MS, 
 and tlie arc SM ; or to the area SmsC contained by the numeric mo- 
 dulus SC, the affymptote C/, and tlxe infinite branch mS of the curve. 
 
 B I N I S.
 
 QliiiWjas 
 
 =r;^-K2eKig2J= 
 
 sasBP^ 
 
 LIST OF BOOKS, 
 
 QJJOTED OR CONSULTED, to ELUCIDATE 
 
 THE 
 
 LIFE AND WRITINGS, 
 
 JOHN NAPIER 
 
 OF 
 
 M E R C li I S T O N. 
 
 ARCIIIMEDIS Syrucafani Arenarius, 
 &c. Eulocii Afcalonitw in hanc cotn- 
 nient. cum verfione et notis Joh. Wallis. 
 
 Diftionaire Hiftorique et Critique par M. 
 Pierre Bayle. A Rotterdam, 1720. Folio 
 paflim. 
 
 Balcarres' Memoirs. 
 
 Bernoulli Ars conje£landi et tra£latus de fe- 
 riebus infinitis. B.itilea;, 1713. 4to. 
 
 Opera Omnia. Laufannjc ct Genevre. 
 
 1742. 
 
 Biographia Britannica. 
 
 Bofcovich de Cycloide ct Logiftica. 
 
 Arithmetica Logarithmica five logaritlimo- 
 rum cliitiades triginta ; pro num':ris natii- 
 rali ferle creftcntibus ab unitate ad 20.000 
 rt a 90.000 ad 100.000 : quorum ope mul- 
 ta perliciuntur arithmetica problemata et 
 geoinctrica. Hos numeros primus invcnit 
 clariflimjis vir Johannes Neperus Baro 
 Mcrchiftonii : cos aiitem ex cjuldem fen- 
 tentii mutavat, cor^imquc ortum et ufum 
 illuftravit Henricus Driggius, in cclcberri- 
 ma Academia Oxonlcnli Gcometriae Pro- 
 
 feflor Savilianus. Deus nobis ufuram vi- 
 tx dedit et ingenii, tanquam pecuniie nul- 
 la prxftituta die. Londini 1624, Polio. 
 
 Boethius de Arithmetica. 
 
 Caufaboni Epiftolse. 
 
 Chriftophori Clavii Bambergenfis, c Soc. Jc- 
 fu, Opera Mathematica. Moguntiae 161 1. 
 Polio. 
 
 . de Aftrolabio. Vol. III. 
 
 Chambers' Diflionary, 2 vols Folio. 
 
 Craufurd's Peer,«ge of Scotland. 
 
 lives of the officers of State. 
 
 Crugerus Pref. in Praxin Trig. 
 
 Nouveau Diet. Hift. etCrit. pour fervir de fup- 
 plement au Dift. de M. Bayle, par Jaques 
 George dc ChaulPepie. Amllerdam 1756^ 
 Folio paflini. 
 
 Douglas's Peerage of Scotland. 
 
 Duteus inquiry. 
 
 Exercitationcs Gcomctricx. Au<5l. Jacob. 
 Gregory. 1 668. 
 
 Hervarti ah Hohenburgh opera. 1610. 
 
 Matlvcmatical Tables containingCommon, Hy- 
 perbolic and Logiaic logarithms; alfo lines, 
 
 Tangent J
 
 L I 
 
 F 
 
 BOOKS. 
 
 tangents, fecants and verfed fines, both na- 
 tural and logarithmic, &c. to which is pre- 
 lixed a large and original hiltory oi' the dil- 
 {overics andwritingi relating to tliofc fub- 
 jffts f<c. By Charles Hutton.LLD. F. K. S. 
 iind Prof. Math Royal Acad. Woolwich. 
 Lond. 1785. 
 
 Humes Hiltory of the Stuarts. 
 
 Keill de Log.. 
 
 Joannes ICeppleri alioruniqueEpillolx mutux. 
 LipfiX 1718. Folio paflim. 
 
 .< Ephemerldes novx motuuniCoelefli- 
 
 um ab anno 1617. 
 
 I.eyburnes Recreations. Folio. 161^4. 
 
 Lilly's Life. London, 1721. 3vo. 
 
 Moreland Sir Samuel. 
 
 liiftoire des Mathematiqnes par M. Montu- 
 cla de I'Academie Royale des Sciences & 
 Jielles-Leltres de I'rulle. 2 tomes quarto, a 
 Paris 1758. paflim. 
 
 Newtoni Principia. AmlL 1723. 
 
 Phil. Trani'jtt. London. 
 
 Micolai Raymari Urfi Ditlimarfi Fundamen- 
 tum Allronomicum, id eft, nova doctrina (i- 
 mium et triangulorum eaque abiblutiflima 
 et pcrfe(rtiirinia ejufque ulus in allronomi- 
 ca calculatione et obfervatione. Argento- 
 rati. 1588. 4to. 
 
 Vitpe quorundam erudifTimoruni et ilhiftri- 
 univirorum, fcriptoreThoma Smith. Lon-. 
 dini 1707. Couimentariolus de vita et 
 fcriptis D. Henrici Briggii. 
 
 Stifellii Arithmetica Integra. 
 
 Tabulx' Rudolphinx, quibus Anronomicte fci- 
 cntix temporuni ionginquitate coilapla; 
 reftaurat o continetur ; a Phcenice illo af- 
 tronomurum Tychone, ex illuftri et gene- 
 rofa Braiieorum in regno Danije familia o- 
 riundo equlte, &c, &:c. curante Joanni Kepp- 
 Icro. Ulmw, anno 1627. Folio. 
 
 Sir Thomas Urquhart of Cromertie's Triflb- 
 tetras. London, 1650. 4to. 
 
 Traft-=. Edinburgh, 1774. 8vo. 
 
 Voffius de Mathemat. 
 
 Reid's Eflay on the Log. 
 
 Wallifii Opera. 
 
 Oughheds' Clavis Mathematica. Oxford, 
 1677. 8vo. 
 
 Worcefter Marquis of, his Scantlings of mo- 
 dern Inventions. 
 
 Craig's quailraturo of the logarithmic curve. 
 
 Rabdologiie feu numerationis per virgi^las li- 
 bri duo. Edinburgi, 1617. i2mo. 
 
 A plain difcovery of the whole Revelation of 
 St Jolin, &c. by John Napierof Merchillon. 
 Edinburgh, ijyj. 4to. by Andio Hart. 
 
 IMirijiti Logarithmoruni Canonis defcriptio 
 &c, Authore Joanne Nepero Barone IVlcr- 
 chiftonii Edinburgw 1O14 apud And. Hart. 
 
 Iniitatio Nepeirea a Henrico Briggs. MSS. 
 1614. in the Britifli Mufeum. 
 
 Ayfcongles Cat. of the MSS in the Britifli 
 Mufeum. London. 2 vols 4to. 
 
 Pcrault des hommes illuftres. 
 
 Tychonis Brahxi vita, Gaflendo Authore. 
 Pariliis 1O54. 4to. 
 
 Pitifci Trigonometria. 
 
 The lives of the ProfeiTors of Grefliam Col- 
 lege by Jolm Ward, Prof, of Rhetoric in 
 Grelham College. F. R. S. London 1740. 
 Folio. 
 
 Logarithmotechnia five methodus conflruendi 
 Logarithmos nova accurata et facilis ; 
 fcripto antehac tommunicato, anno fc. 
 1667 nonis AuguUi: cui nunc accedit ve- 
 ra quadratuia hyperbnlx et inventio fum- 
 mx logarithmorum, Audlore Nicolas Rltr- 
 catore Holfato, e Societate Regia Londini. 
 1668. 
 
 Developpement Nouveau de la partu elemen- 
 taire des Matiiematiques prile dans toute 
 fon etendiie : par Louis Bertrand. i, tomes 
 Geneve, 1778. 4to. 
 
 Trigonometria Britanni<;a. Goudx 1633. Fol. 
 
 Memoires de Mathematique et de Phyiique da 
 I'annee 1756. tirees des regiflres de I'Aca- 
 demie Royale des Sciences a Amfl^erdam 
 1 768. La trigonometric fpherique reduite 
 a quatre analogies par ^L Pingrc. 
 
 Sherwin's Tables. 1771. 
 
 Encydopedie ou Didlionaire raifonne des fci- 
 cnces des arts et des metiers. Neufchal-- 
 tel, 1765. Fol. paflim. 
 
 Roger! Cotefii Harnionia Menfurarum. Can- 
 tabrigix, 1722. 
 
 Abridgement of the Philofophical Tranfafli- 
 
 ons
 
 LIST 
 
 V 
 
 BOOKS. 
 
 ons by Lowthorp et Motte. 5 vols 410. 
 pafllm. 
 
 Atlicnx Oxonicnfes, by Anthony Wood. 2 
 vols I'olio. London, 1691. 
 
 "Woods Hill, et Ant. Ox<jn. 2 vols Folio. 
 Oxonia, 172.J. 
 
 Tables portativcs de Logarithmes, pubiices 
 a Londres par Gardiner, augmentces ct per- 
 f'ectionecs dans leur dil'podtion par M. t."v\!- 
 let, et corrij'ees avec la plus fcriipuleufe 
 exactitude: contenant les logarithmes des 
 nonibrcs depuis i jufqu'a 101960, les lo- 
 logarithmes des finus et taf)gentes, de ie- 
 coiidc en Iccondc pour les deux premiers 
 degres et de 10 en 10 fecondes pour tous 
 les degres du ipiart dc ccrclc •, precedees 
 d'un precis elementaire iur I'explication et 
 I'ufage des logarithmes et fur leur applica- 
 cation aux calculo d'interets, a la Geome- 
 
 tric-pratique, a I'AftronOTr.ie et a la Navi- 
 gation ; fulvies de pluficurs tables interef^ 
 fantes et d'un dil'cours qui en facilite ruiage. 
 a Paris, 1783. 
 
 Univerfale Trigonometria lineare et logarit- 
 mica da Geminiano RondtUi, Prof, di Mar. 
 nello ftudio di Bologna. Bologna, 1705. 
 
 Philofophical Traniaiftions for the year 1695. 
 A ir.ofl compendious and facile method for 
 conftructing the logarithms, exemplified, 
 and denionltrated from the nature of num- 
 bers, without any regard to the hyperbola, 
 with a fpeedy method for finding the num- 
 ber from the given logarithm ; By £. Hal- 
 ley. 
 
 Trig. Plan, et Spher. Elcm. item de Natura 
 et ArithmeticalogarithmorumtraiSkatusbre- 
 vis. Oxon. 1723. 8vo. 

 
 En^'avp<IJbrdie Life of NAPuai 
 
 I'ljte 1 
 
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