Chapter VI (2)

The Great Appendix of Natural Philosophy both Speculative and Practical. Mathematics. Its Proper Position not among the Substantial Sciences, but in their Appendix. Mathematics divided into Pure and Mixed

IT WAS well observed by Aristotle, that physics and mathematics produce practice, or mechanics; therefore as we have treated both the speculative and practical part of the doctrine of nature we should also consider mathematics as an auxiliary science to both, which being revived into philosophy, comes in as a third part after physics and metaphysics. But upon due recollection, if we designed it as a substantial and principal science, it were more agreeable to method and the nature of the thing to make it a part of metaphysics. For quantity, the subject of mathematics applied to matter, is as the dose of nature, and productive of numerous effects in natural things, and therefore ought to be reckoned among essential forms. And so much did the power of figures and numbers prevail with the ancients, that Democritus chiefly placed the principles of the variety of things in the figures of their atoms; and Pythagoras asserted that the nature of things consisted of numbers. Thus much is true, that of natural forms, such as we understand them, quantity is the most abstracted and separable from matter; and for this reason it has been more carefully cultivated and examined into by mankind than any other forms, which are all of them more immersed in matter. For, as to the great disadvantage of the sciences, it is natural for men’s minds to delight more in the open fields of generals, than in the inclosures of particulars, nothing is found more agreeable than mathematics, which fully gratifies this appetite of expatiating and ranging at large. But as we regard not only truth and order, but also the benefits and advantages of mankind, it seems best, since mathematics is of great use in physics, metaphysics, mechanics and magics, to make it an appendage or auxiliary to them all. And this we are in some measure obliged to do, from the fondness and towering notions of mathematicians, who would have their science preside over physics. It is a strange fatality, that mathematics and logic, which ought to be but handmaids to physics, should boast their certainty before it, and even exercise dominion against it. But the place and dignity of this 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 regard to progressions, whose use in physics is very considerable. Neither is algebra brought to perfection. As for the Pythagorical 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 sufficiently comprehended, clearly demonstrated, nor dexterously fitted for use. And of this kind are perspective, music, astronomy, cosmography, architecture, and mechanics. In mixed mathematics 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 sylvæ,”—

the voice of nature will cry it up, though the voice of man should cry it down; and as Alexander Borgia was wont to say of the expedition of the French against Naples, that they came with chalk in their hands to mark up their lodgings, and not with weapons to fight, so we prefer that entry of truth which comes peaceably, when the minds of men capable of lodging so great a guest are signed as it were with chalk, than that which comes with pugnacity, and forces its way by contentions and controversies. Wherefore, having gone through the two parts of philosophy that relate to God and to Nature, we come to the third, which is man himself.