VOL. XXI.] PHILOSOPHICAL TRANSACTIONS. 411 



musicians. To this he gives a new Latin translation, with copious notes, and 

 clear explanations of what might seem difficult in the Greek music. To this 

 he subjoins an excellent treatise of his own, comparing the ancient Greek music 

 with that of the present age; by which that which was before admired rather 

 than understood is now rendered very intelligible, according to the language of 

 modern music. Next to this is the Commentary of Porphyry on a great part of 

 Ptolemy's Harmonics, with like notes and necessary emendations as the former. 

 Then follow the Harmonics of Manuel Bryennius, in Greek and Latin, with 

 notes and necessary emendations as the rest. So that now we have all the ancient 

 Greek musicians known to be extant, published in Greek and Latin; Marcus 

 Meibomius having formerly published several of them in the year l652, and the 

 remaining three being now added. 



The next piece is Archimedes's Arenarius ; of this we had a Greek edition 

 of Hervagius, published at Basil in the year 1544; which seems to be done by 

 Hervagius with great care and fidelity, but out of a very faulty manuscript copy. 

 To this is subjoined that of Archimedes, called KJiiAa MtTfuo-i?, or Dimensio 

 Circulij which had been formerly published in Greek with other works of Archi- 

 medes, in the Basil edition, by Hervagius. 



To this is now added the Commentary of Eutocius, by which it is shown how 

 troublesome it was to perform the arithmetical operations of division and extrac- 

 tion of roots, and other intricate operations, before the introduction of the Indian 

 algorism, or calculation by the numeral figures now in use, of which Archimedes, 

 in his Arenarius, gives us the true foundation as to the economy of numbers, 

 but without the notation now in use. 



After these pieces of Archimedes and Eutocius, follows a treatise of Aris- 

 tarchus Samius, De Magnitudinibus et Distantiis Soliset Lunae, first published 

 by Dr. Wallis out of some manuscript copies in the year 1688, and now re- 

 printed with the Latin translation and annotations of Commandine and of his 

 own. 



To this is subjoined in Greek and Latin, a fragment of the second book of 

 Pappus Alexandrinus's Mathematic Collections. The Latin translation of which 

 author, published by Commandine, begins at the third book, the two former 

 being wanting. But a good part of the second book is now published in Greek 

 and Latin, by which we may judge of the contents of what is lost ; and that 

 the loss is not great, as giving an account of the arithmetical operations then in 

 use ; which are now performed with much more advantage by the algorithm or 

 numeral figures now used. 



After this preservation and restitution of these ancient Greek authors follows 

 a collection of divers letters, relating to mathematical affairs, which have for- 



8 G 2 



