387 
quilibet, e divisione eliciendus, per solam subtractionem, sine 
tediosi & lubric&é Multiplicationis, atque Divisionis opera- 
tione, etiam ab eo, qui Arithmetices non admodum sit gnarus, 
exacte, celeriter & nullo negotio invenitur. E museo Ioannis 
Georgii Herwart ab Hohenburg...Monachii Bavariarum...Anno 
Christi, M.DC.X.”, and the book is a very large and thick 
folio. It contains a multiplication table up to 1000 x 1000, 
the thousand multiples of any one number being given on the 
same page; and there is an introduction of seven pages, in 
which the use of the table in multiplying numbers containing 
more than three figures, and in the solution of spherical 
triangles, is explained. 
Very little information about the work is to be obtained 
from the mathematical bibliographers and historians. Heil- 
bronner (Hist. Math. 1742, p. 801) gives the title not quite 
correctly, and adds “Docet in his tabulis sine abaco multiplica- 
tionem atque divisionem perficere.” Kastner (Gesch. der Math. 
1796—1800, t. iii. p. 8) quotes the title from Heilbronner and 
‘his remark, and adds that the latter could not have known 
Herwart’s method, or he would have described it. He remem- 
bers to have read somewhere that the book contained a number 
of tables of products, arranged by factors, like a great multipli- 
cation table. Scheibel (Hinl. zur math. Buch. 1775, t. 1. 
p- 417) gives the title-page correctly, and explains the method 
of using the table when the number of figures in the multiplier 
or multiplicand exceeds three, and concludes with the remark, 
“So viel von diesem ungeheuren Folianten, den man bloss zur 
Curiositat und seiner Seltenheit wegen, in einer mathema- 
tischen Biichersammlung aufbewahret.” Montucla (Hist. des 
Math. t. ii. p. 13) gives a description of the mode of using the 
table, remarking that but for the invention of logarithms it 
might have been of use to calculators, supposing the labour of 
searching for the products in so large a folio not to be more 
fatiguing than the direct performance of the work. Murhard 
