ON MATHEMATICAL TABLES. 45 



his tables not only to this day remain unsuperscded and the ultimate authori- 

 ties, but also formed the data whereby Vlacq calculated his logarithmic 

 canon. Pitiscus says that for twelve years Eheticus constantly had some com- 

 puters at work (duodecim totos annos semper aliquot Logistas aluit) ; and how 

 much labour and expense on his part would have been wasted but for the 

 zeal of Pitisciis is painful to contemplate ; as it was, it is matter of regret 

 that llheticus did not live to see the publication of either of his canons, 

 the first of which appeared twenty years, and the other thirty-seven years 

 after his death. It was Pitiscus's intention to add Eheticus's minute-canon 

 of tangents and secants ; but they laboured under the same defect as those in 

 the (uncorrected) ' Opus Palatinum,' and on this account ho was dissuaded 

 from so doing by Adrianus Romanus. The matter spoken of above as 

 [T. III. and IV.] was due to Pitiscus himself, and was introduced at the 

 advice of the same mathematician. 



The enormous work undertaken by Eheticus needs no eulogy ; and the 

 earnestness and love of accuracy displayed by Pitiscus, not only rendered 

 apparent by his acts but also evident in the prefaces to his several works, 

 will always render his an honoured name in science. 



The present work is exceedingly rare ; and the copy wc have examined is 

 in the library of the Greenwich Observatory. It, the ' Opus Palatinum,' 

 and Vlacq's ' Arithmetica Logarithmica,' 1628, and ' Trigonometria Artifici- 

 alis,' 1633, may be said to be the four fundamental tables of the mathemati- 

 cal sciences. 



Gernerth (in the work cited under Rhexictis, 1596, supra) has given a 

 list of 88 errors that he detected in the first 7 or 8 places of the canon of 

 sines; he detected altogether 110; but 22 he states were given by Vega 

 in his ' Logarithmisch-trigonometrische .... Tafeln und Pormeln,' Vienna, 

 1783 : this was Vega's first publication of tables ; and we have not seen the 

 work. 



Grienberger, 1630. Sines, tangents, and secants, to 5 places, for every 

 minute from 0° to 45° (with foot entries also ; but the table is only half a 

 complete canon, as e.g. sin 50° could not be taken out from it). There are five 

 more figures added to the sines, but separated from them by a point (this is 

 not a true decimal point, as is evident from the rest of the work, where no 

 trace of decimals occurs), the object the author had in view in adding them 

 being that when the sines had to be multiplied by large numbers, the re- 

 sults might still be correct to the last unit (radius 100,000). Grienberger 

 stated that more than 35 years before (about 1595) he had calculated a 

 canon of sines to 1 6 places, and made considerable progress with the secants 

 when the ' Opus Palatinum ' appeared and caused him to lay aside his work. 

 This he regretted exceedingly at the time of writing the present work, as he 

 was not able to add the five extra figures to the tangents and secants, which 

 he had transferred from his MS. in the case of the sines. The ' Opus Pala- 

 tinum' contained enough figures; but some of them were doubtful, and he 

 wished no doubt to attach to any part of his table. The book is a duodecimo 

 volume, and would scarcely have been noticed here, but from the f;ict of j)art 

 of it having been the result of an original calculation. Napier's bones are 

 mentioned, but not logarithms. The preface contains Grienberger's 39-figure 

 value of TT (see ' Messenger of Mathematics,' July 1873) ; and it was in con- 

 nexion therewith that we sought the work out, and learnt with some surprise 

 of Grienberger's incomplete and unpublished calculations. The copy we 

 examined is in the British Museum. 



Massaloup, 1847, T. I. The first five hundred multiples of the sines and 



