- 



T.UiLK. 



and 10,000,000 are historical facto of notoriety. Our question is, 

 what tables were first printed I On the books which Hegiomoutanus 

 actually printed, out of the long lint* of those which he published and 

 intended to publish (as set forth in his own ' Index Operum," &c., 

 |irint,d ,-it NurnWrgby himself), hid historians. DopjiclmayiM-, l)e Murr, 

 Weidlcr, &c., are either not very clear, or somewhat nt variance. In 

 the vague manner in which liooks and their contents are frequently 

 described by professedly mathematical writers, a good resource* is 

 often found in the catalogues of genn ul bibliographers. 



The ' Tabuho Dtrectionetn Profectionumque ' of Kegiomoutanus were 

 published by himself at Nuruberg, without date, probably about (1475), 

 and were reprinUtl at Venice in (1485). We cannot ascertain that 

 either of these contained tables of sines. But Hain (' Repert. Bibliogr.'), 

 who gives their titles, gives that of the next edition, Augsburg, Khr. 

 Katdolt (1490), 4to, in a fuller manner : from which it appears that 

 i!i.-:,- is appended to it a table of sines to minutes, in words which 

 would imply that Kegiomontanus bad not given such a table in the 

 former edition : they arc, ' Tabella Sinus recti : per gradus et eingula 

 minuta divisa. Ad Tabulas Directionum Mag. Job. de liegiouionte 

 necessarias.' But from the description it is clear that this table does 

 not belong t the work, since it follows even the printer's insignia. 

 And Hain also met with it as a separate work ; being, as appears from 

 his description of the lineation, pages, &c., absolutely the same as that 

 which was appended to the Tabulfc Profectionum. Accordingly, 

 until something earlier or more definite is produced, we must say that 

 the first known printed table of sines is an anonymous table, to 

 minutes, in quarto, without date, but before 1500, stated (with nece- 

 taricu when it ought to be neceuaria) to be necessary to the tables of 

 Kegiomontanus, and implying that sines had not then been printed 

 with those tables. From the next-mentioned edition of the Tabulae 

 Directionum (this we have seen) we should suppose that these tables 

 were to a radius of 600,000, as in that edition, which is of Venice, 

 1504. 4to. In it we find a minute-table of sines, headed ' Incipit tabella 

 sinus recti,' and with a column containing differences for ten seconds. 

 Delambre and others mention Kegiomontanus as having given the first 

 table of tangents in this work under the name of tabula fircunda. It 

 is in the edition of 1 504, and was reprinted by Gemma Frisius in his 

 book' De Kadio Astronomico," Antwerp, 1545. It is to degrees only, 

 and to a radius 100,000 ; and is a table of cotangents, not of tangents. 

 The Tubingen edition, 1550, also distinguishes the table of sines as 

 an addition, in the title-page. The radius is now 60,000 ; but, by an 

 oversight, the differences to ten seconds are entered from some table 

 with a radius of 6,000,000, from which the table of 1550 was probably 

 cut down. There is no^abitla fcecunda. Delambre mentions an edition 

 of the work, edited by Gauricus, in (1524), as containing a table of 

 sines to every ten minutes : of this we can find nothing. 



As yet we have no sines calculated to the now ordinary radius of 

 10,000 &c. Of these, the earliest we have seen (and we tind no earlier 

 ones mentioned), are those of Peter Apian in the ' Introductio Geo- 

 graphica.' 4c., Ingoldstadt, 1533, folio. They are minute-tables to a 

 radius of 100,000, and were reprinted the next year in the same author's 

 ' Instrumeutum primi Mobilis.' Nurnberg, 1534, folio. Apian states 

 that they are of his own calculation, and this is to us a strong presump- 

 tion that no such tables had been previously printed ; for Apian was a 

 great reprinter of the writings of others at his own press, and very 

 unlikely to have re-calculated any table which he knew to exist already. 

 The statement that the work of Kegiomontanus on triangles, Nurn- 

 berg, 1 533, folio, contains tables of sines, is incorrect : we know it from 

 examination of two perfect copies. We can point out how the mistake 

 arose. Lalande (Bibl. Astron.) says that the first edition of the work, 

 Basle, I,i36, has in the title-page " una cum tabulis sinuum." Now the 

 fact is that Lalande, who had only seen the second edition (Bade, no 

 date, known to be of 1561), which does contain tables of siries, took the 

 liberty of presuming that the first edition was the same in contents, 

 title, and place ; in all of which he was wrong, and in the date also. 



In 1''42 Rheticus, the most laborious of all the table-computers, 

 made his first appearance as the editor of a work of Copernicus, ' De 

 Lateribus et Angulis Triangulorum.' &c., Wittemberg, 4to (Wcidler 

 and Kastner). This contains a minute-table of sines to a radius of 

 1 0,000,00(1, being the first- published seven-figure table : the copy in 

 the Libri sale of April, 18ol, is the only one we ever saw. The table 

 which appeared in the following year, in the great work of Copernicus 

 [COPERNICUS, in Bioo. Div.], is an abridgment of the preceding; 

 going only to every ten minutes, and to a radius of 100,000. 



In 1541 appeared one of the tables which have obtained most cele- 

 brity : being the ' Tractatus Geo. Purbachii super propositioues 

 Ptolemsei de Sinubus et Chordis, item compositio Tabularum Sinuum 

 per Joannem de Regiomonte. AdjeeUc sunt TabuUo Sinuum duplices 

 per eundem Regiomoutanum,' Nuruberg (1541), folio (Kastner, &c.). 

 The two tables of sines are both minute-tables, with radii of 6,000,000 

 and of 10,000,000. The table of tangents to every degree is repeated 

 again under the name of tabula fiecunda. In a mixture of tracts by 

 Kegiomontanus, Walther, Schoner, and Purbach, headed ' Scripta . . . 

 <!> Torqueto, Astrolabio Armillari, ..." Nurnberg, 1544, 4to, is a 



* It ould be better if wo knew precisely irhrn it is good. Not to believe 

 more than half Is > rcry proper caution : but there arises the old difficulty, 



Which b.lf r 



TABLE. 



table by Purbach, called Tabula Gnomonica, which is a table of tan 

 genU to the radius 1200, giving the angle to each unit of the 1200. 

 Thin table is repeated in Gemma Friaius, ' De Radio Astronomico,' 

 Antwerp, 1545, 4to, already mentioned. 



RheticiiH, in the meanwhile, was pursuing the route of analogy, which 

 suggested to him the formation of a table giving all the ratios which 

 exist between the sides of aright-angled triangle; by which In- 

 led to the invention of what were afterwards called secants, to the 

 completion of the trigonometrical canon, and to its arrangement in the 

 (Will which it has ever since preserved. His rights in this matter 

 have long been forgotten ; and it is only recently that the work 

 which established them haa received any notice in modem times. (See 

 the Notices of the Astron. Soc., vol. vi., p. 213, and Phil. Mag., 

 June, 1845.) In 1551, the year following that in which he was placed 

 in the ' Index ' as a forbidden author, he published his ' Canon Doc- 

 triiiic Triangulorum,' Leipzig, 4to. This is a complete canon to every 

 ten minutes, and to a radius of 10,000,000 (or, as we should now say, 

 to seven decimals) with differences, so arranged that the matters con- 

 nected with each angle also belong to its complement, in the manner so 

 familiar to those who can use any modern table. This arrangement 

 may be called temi-ijua<lrantal, as opposed to the older i/itail 

 arrangement in which the sines are carried direct from to 90. 

 Accordingly, the page of Rheticus has both a head and foot description, 

 as in modern tables. So completely is he bent on the idea of a register 

 of the proportions of right-angled triangles, that he rejects the use of 

 the word sine. In the place of the sine and cosine he has the perpen- 

 dicular and base to an hypothenuse of 10,000,000; for what were 

 afterwards called the tangent and secant he has the perpendicular and 

 hypothenuse to a base of 10,000,000 ; for the cotangent and cosecant he 

 haa the base and hypothenuse to a perpendicular of 10,000,000. The 

 same description is adopted in his larger work, of which we shall pre- 

 sently speak. To the smaller work is appended a dialogue, which 

 introduced Rheticus to his future editor ; for Valentine Otho was so 

 struck by it, that he went to Hungary to obtain information on the 

 subject from the author. Otho relates that in the first interview, 

 when he had hardly stated his purpose, Rheticus interrupted him 

 with, " You are just as old as I was when I went on the same errand 

 to Copernicus." 



In 1554 Erasmus Reinhold (who had been the colleague of Rheticus * 

 in teaching mathematics at Wittemberg) published the ' Primus Liber 

 Tabularum Directionum,' Tubingen, 4to. In this work, for the first 

 time, occurs a canon fcecundus (not yet called a table of tangents) car- 

 ried to every minute. Both sines and tangents were computed to a 

 radius of 10,000,000, and have differences. By the complemental 

 degrees at the bottom of the pages, it appears that cosines and cotan- 

 gents were intended. This work of Reinhold, though founded upon 

 Regiomontanus, must not be confounded with his professed edition 

 of the ' Tabula; Directionum ' of Regiomontanus himself, which had 

 tangents only to every degree, and was printed several times, the last 

 edition being in (1606). (\Ve have not thought it worth while to cata- 

 logue all reprints.) 



In (1558) (Delambre) Maurolycus published his edition of Theodo- 

 sius, Menelaua, &c. (Messina, 4to), containing the three tables, that of 

 sines, the tabula fcrcunda, and the tabula, bcnejica (as he called the pre- 

 sent table of secants). This table goes only to degrees (except that 

 tangents and secants are given for 15, 30, 45, 55, and 59 minutes of the 

 last degree of the quadrant), and is to a radius of 100,000. Delambre, 

 &c., suppose that these are the first tables of secants which were pub- 

 lished, and they accordingly attribute the invention to Maurolycus. 

 But we have seen that it is due to Rheticus; and Fiuck (presently 

 mentioned), who lived close to these times, states expressly that Mauro- 

 lycus borrowed this table from Rheticus. 



In 1502 a pupil of Rheticus published a table of sines to every 

 minute, and to a radius of 10,000,000, with differences for one second. 

 This was Samuel Eisenmenger (or Siderocrates, as he wrote himself), 

 in his ' Libellus Geographicus,' Tubingen, 4to. And there was, as we 

 find stated in various quarters, a table of sines in the work on dialling 

 of Hermann Witekind, ' Conformatio Horologiorum,' of which the first 

 edition is said to be of Heidelberg (1576), 4to. Blundeville says they 

 are to a radius of 100,000. 



The first complete canon to every minute (that of Rheticus in 1551 

 being to every ten minutes) was Vieta's ' Canon Mathernaticus, seu ad 

 Triangula, cum Adpendicibus,' Lutetia;, apud Johannem Mettayer, &c., 

 1579 ; to which is annexed, with a new title-page, ' Francisci Vietse i 

 universalium Inspectionum ad Canonem Matheuiaticum liber singularis, 

 Lutetiie,' &c., as before. 



This same book, from the same types, is also found with another 

 title-page, as follows : ' Francisci Viefaei opera mathematica, in quibuu 



* Reinhold taught the higher branches, and Rheticus the lower. It is very 

 illustrative of the neglect into which the prohibition (with other circumstances 

 afterwards noted) caused the writings of Kheticus to fall, that \\eicllcr himself 

 of the university of Wittemberg, writing and printing his History of Astronomy 

 there, giving minutely the dates of llheticus's degrees from the matricuta or 

 register, and stating that from the time when he and Kcinhold were colleagues 

 it had always been customary to bave two teachers of mathematics is as ill- 

 informed as any one about the writings of Rheticus, and in particular knows 

 nothing of the publication of 1591, of which we may therefore be pretty .sure 

 there was not a copy in the library. 



