: 



T.Vr.LK. 



TAliLK. 





Delambrc propose* to call them .Vaptrian logarithms, and to restrict 

 the term kyprrtnilic to the modern \ajieria H or logarithms : but 

 custom has refused. 



The reader may have some curiosity to see a specimen of the 

 princrpt : we therefore subjoin three lines from the page having (Jr. :>7 

 at tin- top and Or. 52 at the bottom. The form is semiquadrantal, 

 and the it if treat la is the logarithm of the tangent. 



This work of Napier was reprinted by Baron Maseres, in vol. vi. of 

 the ' Scriptures Logarithmiei.' 



(1816.) Keprint of the above, by Edward Wright, to one figure less, 

 with Napier's explanation translated into English, and preface by 

 Briggs, London. 



(1617.) Briggs, ' Logarithmorum Chilias Prima,' London. This is 

 the first publication of logarithms on Brigga's system. 



(1'ilS.) Benjamin Ursinus, ' Cursus Mathematicus,' Cologne, con- 

 tains Napier's Logarithms. For his ' Magnus Canon,' see 1624. 



(1619.) John Speidell, ' New Logarithms.' A new arrangement of 

 Napier's, but giving sines, tangents, and secants, with numbers also to 

 1000. The.e ' New Logarithmes' are the first modern \aperinn, or 



..rrbolic logarithms. The second edition was in 1620, not 1627, as 

 we stated (from others) before we had seen it The reason of the 

 mistake is that the ' Briefe Treatise of Sphicricall Triangles,' which is 

 frequently prefixed, has 1627 on its title-page. 



Taking decimals it stands thus : If m : n be the sine of an angle , 

 and if A represent the logarithm to the base , the jtyitrcs of the 

 Naperian logarithm are found in 



AH Am. 

 Thus, the sine of 19 38' is '336, very nearly. And we have 



A 1000 = 6-9077552. .. 



-171111 . . . 

 1 u!niii441 Napier has 10906448. 



The figures of Speidell's logarithmic sine are found in 

 10+Am AH 



thus for 19 38' he has 890936. But he leaves the 10 out of all the 

 secants and the last half of the tangents. His logarithms of numbers, 

 (1) 1000, are modem hyperbolic to six decimals, as we should now 

 gay, but without the decimal point; thus at 770 he has 6646388 not 

 6.646388. To each logarithm he gives its difference, its arithmetical com- 

 plement, and the halves of all three. Also an additional column which 

 shows that he means to use his table in calculation by feet, inches, and 

 quarters. Thus the number 775 has 16.1.3 opposite to it, there being 

 775 quarter inches in 16 feet 1 inch 3 quarters. At the bottom of each 

 page he puts the logarithm of 100 and of 1000, for help in decimal 

 fractions. 



Speidell, as we have seen, first published in (1619) ; Baron Maseres 

 reprinted from the "tenth impression." dated (1628); there was an 

 edition in (1627); Hutton mentions the seventh, dated (Ifi24); we 

 have the fifth, dated 1623, and the sixth, dated 1624 ; the Royal 

 Society has one of 1623 ; Murhard gives the third impression, of (1621), 

 and we have the second, dated 1620. In his " briefe treatise " above 

 mentioned, Speidell mentions, and naturally complains of, those * who 

 had printed his work without an atom of alteration, and yet dispraised 

 or undervalued it in their prefaces for want of alterations which them- 

 selves either could not or would not make. This he attributes to his 

 not having been at Oxford or Cambridge. t Having kept our eye on 

 thin work until we have obtained four copies of it, thouph the British 

 Museum? does not possess one, without ever finding the smallest trace 



To them he speaks as follows, in the introduction : 



"To theM. C. Z. 

 If that thou canst amend it, 

 So shall the Arte increase : 

 If thou canst not : commend it, 

 Else, prcethee hould thy peace." 



t " Yet to satisflc in part the learned, that I can piuc a reason for what 1 

 dor, I will >ct downc the making of these 3. last Theorems, whereby they may 

 (If no they please) suppose I can doe as ranch for the rent, anil whether some of 

 them doc or no, I pane not greatly, for that they arc sorry I can doe BO well, 

 not hailing wcnc one of the Vniurraities " (p. 27). 



J The Brltiih Museum ii well supplied with mathematical works of the 

 period : and the deficiency Illustrates what < shall say on the decadence of 

 J. Bpcldcll and his works. It is very much to be regretted that the 

 did not purchase Dr. l[iitton> library. The niatt'-r was in discussion, and 

 alrormt In negotiation ; but things were prevented from going further by 

 Sir Joxph Hanks. Hutton distinctly declared, both at the time of the tale and 

 after, that hla "old Implacable crrn-jr" hud prevented the Museum from 



of any reproduction by another hand, we permit ourselves to doubt 

 Speidell's assertion about the reprints : and the more readil . 

 finding out the reasons for suspecting him of unfairness of which we 

 -li,,ll presently speak. 



Whether for his own reason or not, Speidell's name was very little 

 * The Continental writers rarely mention him; Walhs knew 

 nothing of him ; and even his own son, Euclid Speidell, when he pub- 

 lished his ' Logarithmotechnia,' in 1 688, had no accurate information 

 on his father's writings ; for he says, " I do find my father printed 

 several sorts of logarithms, but at last concluded tli uial or 



Briggs's logarithm- -.-. re the best sort for a standard logarithm, and 

 did also print the same several ways." This must have been m 

 mistaken tradition, arising from Speidell's not having printed the same 

 logarithms as Napier : we may safely say he did not print any decimal 

 logarithms. In addition to this testimony a< to the rapid spread of 

 logarithms in England which Speidell's circulation give-, we m 

 that their advantages were immediately seen by the practical mathe- 

 maticians. Aaron Rathborne, in his ' Surveyor,' London, 1616, recom- 

 mends the use of the " tables and more than admirable invention of 

 logarithmes by that divine and noble writer the Lord Marchiston, whoes 

 name and honour will never out." 



It is to be noticed that two different tables of common logarithms 

 accompanied Speidell's sines, &c. Our second, fifth, and one of two 

 copies of the sixth, have, through (1) 1000, the logarithm and its c-omph-- 

 ment with the common difference between them, and, by the 

 them, the semi-logarithm and semi-complement, with (A./V common 

 difference between them. But our other copy of the sixth imp: 

 has nothing but plain number and logarithm, without even the differ- 

 ences. And the larger table has, tfp to 960, an argument in sh 

 pence, and farthings. 



Thus far we had got in the revision of -our former article, when we 

 took it into our heads to compare the four copies before us, and, ex- 

 cepting only the new table of common logarithms in one of t ! 

 copies, we found them to be all from the same type, even t.. the very 

 title-pages. It is true that one has ' The 2. Inpression. 1 021),' and 

 another ' The 6. Inpression. 1624," &c. ; but as much as ' The In- 

 pression. 162 ' is from the same type in all. We cannot be deceived 

 here, though those who are not used to such comparisons may think it 

 possible. The inferior printing of that period abounds in badly 

 formed, ill ranged, and blurred letters : and every instance of mal- 

 f i filiation is common to all our three cases. Hyphens, in particidar, 

 are very good tests: and one in " Ho-nourable " which hap]' 

 slant upwards in all three, and another in "Play-House" which hap- 

 pens to slant downwards, are hardly possible coincidences. Either tin- 

 type was kept standing, and each year's sale was called an im|n 

 or the title-page was several times fraudulently set forward in date 

 after a number of copies had been taken off. There is no printer's 

 name ; and that the type should have been allowed to stand for many 

 years is very unlikely, though we must see that the occuirene. 

 inferior table of common logarithms in the late editions is in 

 of the supposition. It seems to favour the hypothesis that the f < >rms 

 of some of the last pages got broken up by accident, and that the 

 inferior table was set up to replace them. 



1619. Napier. ' Mirifici Logarithmorum Canonis Constructio,' K.lin- 

 burgh, edited by Napier's son. 



1620. Reprint of Napier, both the ' Descriptio ' and the 'Constructio,' 

 at Lyons, by Bartholomew Vincent, bookseller. 



(1620). Guuter, ' Canon of Triangles,' first trigonometrical canon 

 with Briggs's ' Logarithms.' We have not seen the first edition of 

 this canon, which is semi-quadrantal, (!') 45" to seven decimals, and 

 0(1)1000 to eight. It was certainly first published in liiju. 

 and Wingate, in 1624, states it to be of Qunter's own calculation, and 

 acknowledges it as the source of his own reprint. A year or tw . 

 wards, the work on the cross-staff was published, to which this canon 

 was attached : and in future editions the two always went tot 

 The second edition of both was in 1636 ; the fourth (edited by Henry 

 Bond, who perhaps edited the third, of which we know- nothing) in 

 (1662) ; the logarithms of numbers being 0(1) 10,000 to seven decimals. 

 Quitter is entitled to rank as one of the primary calculators of 

 logarithms, with Napier, Briggs, John Speidell, and Vlacq. 



(1620). J. B. [Justus Byrgius], ' Arithmetische und Geoinetrische 

 progresse Tabulen,' Prague. This is the title given by Montuela, and 

 the history of the book is as follows : Kepler had stated that Byrge 

 had invented the very same logarithms as Napier many years i 

 the latter published anything on the subject. And Kramer, author of 

 a German work on perspective, Cassel (1630), says that his brother-in- 

 law anil teacher, Justus Byrgius, had. twenty years before that time, 

 made a table of progressions with differences of 1", > M> nine 



figures, which he had published without text at Prague in 162". This 

 announcement obtained no notice, until Kiistner informed Montuela 



making the purchase, and that snch was the frequent opinion of those who 

 thought about such things. (Bruce, ' Life of Hutton,' Newcastle, 1833). Thus 

 the finest set of mathematical tables ever collected in England wa- di-pi isl. 



* There is not now a copy of any edition in the llritish Museum. Hutton 

 happens to say tliat thirei* in the seventh Imj lie had 



before him) a table of logarithms of numbers. In th" ' Encycl. liiii.' tin- i< 

 translated into an as-eriiim th:M tin- logarithms of numbers were not added 

 until the seventh impression. 



