ADVANCEMENT OF LEARNING 103 



science is a secondary consideration with regard to the thing 

 itself. 



Mathematics is either pure or mixed. To the pure belong 

 the sciences employed about quantity, wholly abstracted from 

 matter and physical axioms. This has two parts geometry 

 and arithmetic; the one regarding continued, and the other 

 discrete quantity. These two sciences have been cultivated 

 with very great subtilty and application ; but in plain geometry 

 there has nothing considerable been added to the labors of 

 Euclid, though he lived many ages since. The doctrine of 

 solids has not been prosecuted and extended equal to its use 

 and excellency, neither by the ancients nor the moderns ; and 

 in arithmetic there is still wanting a sufficient variety of short 

 and commodious methods of calculation, especially with re- 

 gard to progressions, whose use in physics is very considerable. 

 Neither is algebra brought to perfection. As for the Pytha- 

 gorical and mystical arithmetic, which began to be recovered 

 from Proclus, and certain remains of Euclid, it is a speculative 

 excursion, the mind having this misfortune, that when it proves 

 unequal to solid and useful things, it spends itself upon such as 

 are unprofitable. 



Mixed mathematics has for its subject axioms and the parts 

 of physics, and considers quantity so far as may be assisting to 

 illustrate, demonstrate, and actuate those ; for without the help 

 of mathematics many parts of nature could neither be sufficient- 

 ly comprehended, clearly demonstrated, nor dexterously fitted 

 for use. And of this kind are perspective, music, astronomy, 

 cosmography, architecture, and mechanics. In mixed mathe- 

 matics we at present find no entire parts deficient, but foretell 

 there will be many found hereafter, if men are not wanting to 

 themselves ; for if physics be daily improving, and drawing out 

 new axioms, it will continually be wanting fresh assistances 

 from mathematics; so that the parts of mixed mathematics 

 must gradually grow more numerous. 



We have now gone through the physical sciences, and 

 marked out the waste ground in them. . If, however, we have 

 departed from the ancient and received opinions, and arrayed 

 opponents against us, we have not affected contradiction, and 

 therefore will not enter into the lists of contention. If we have 

 spoken the truth, 



" Non canimus surdis; respondent omnia sylvae"<* 



