CHAP. VI.] PURE AND MIXED MATHEMATICS. 149 



perfection. As for the Pjthagorical 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 sjiends itself upon such as are unpro- 

 fitable. 



Mixed mathematics has for its subject axioms and the 

 parts of physics, and considers quantity so far as may bo 

 assisting to illustrate, demonstrate, and actuate those; for 

 without the help of mathematics many parts of nature could 

 neither be sufficiently comprehended, clearly demonstrated, 

 nor dexterously fitted for use. And of this kind are per- 

 spective, music, astronomy, cosmography, architecture, and 

 mechanics. In mixed mathematics we at present find no 

 entire parts deficient, but toretell there will ])e many found 

 hereafter, if men are not wanting to themselves; for if phy- 

 sics 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, anc 

 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 contra- 

 diction, and therefore will not enter into the lists of conten- 

 tion. If we have spoken the truth, 



" Non canimus surdis ; respondent omnia sylvse,"' — 

 the voice of nature will cry it up, though the voice ot 

 man should cry it down ; and as Alexander Borgia was w(mt 

 to say of the expedition of the French against Naples, that 

 they came with chalk in their hands to mark up their lodg- 



varsjB," in two volumes 4to., printed at Halle in the year 1715; or for a 

 more cursory view, Father Castel's " Math^malique Universelle," pub- 

 lished in the j'^ear 1731 ; but for the history of mathematics, see Vossius 

 " De Universae Matheseos Natura et Constitutione ;" the *' Almagest" of 

 Kicciolus; Morhof's " Polyhist. Mathemat. ;" andWolfius's "Commen- 

 tatio de Scriptis Mathematicis," at the end oi the second volume of his 

 *• Elementa Matheseos CJniversse;" Montucla's "Hist. Math.;" and De 

 la Croix's " Analysis ol Infinites." £,d. 



^ He ought to have said Irom larablicus. Proclus was, like himself, 

 totally ignorant eveij of tJlQ Uttle mathematical learning extant in \\\t 



day. I'd, * Virg. Edoiiues, ?. §, 



