Fifthly, that compounded names, which are defined one way in some one part of philosophy, may in another part of the same be otherwise defined; as a parabola and an hyperbole have one definition in geometry, and another in rhetoric; for definitions are instituted and serve for the understanding of the doctrine which is treated of. And, therefore, as in one part of philosophy, a definition may have in it some one fit name for the more brief explanation of some proposition in geometry; so it may have the same liberty in other parts of philosophy; for the use of names is particular (even where many agree to the settling of them) and arbitrary.
Sixthly, that no name can be defined by any one word; because no one word is sufficient for the resolving of one or more words.
Seventhly, that a defined name ought not to be repeated in the definition. For a defined name is the whole compound, and a definition is the resolution of that compound into parts; but no total can be part of itself.
Nature of a demonstration.
16. Any two definitions, that may be compounded into a syllogism, produce a conclusion; which, because it is derived from principles, that is, from definitions, is said to be demonstrated; and the derivation or composition itself is called a demonstration. In like manner, if a syllogism be made of two propositions, whereof one is a definition, the other a demonstrated conclusion, or neither of them is a definition, but both formerly demonstrated, that syllogism is also called a demonstration, and so successively. The definition therefore of a demonstration is this, a demonstration is a syllogism, or series of syllogisms, derived and continued, from the definitions of names, to the last conclusion. And from hence it may be understood, that all true ratiocination, which taketh its beginning from true principles, produceth science, and is true demonstration. For as for the original of the name, although that, which the Greeks called ἀποδέιξις, and the Latins demonstratio, was understood by them for that sort only of ratiocination, in which, by the describing of certain lines and figures, they placed the thing they were to prove, as it were before men's eyes, which is properly ἀποδεικνύειν, or to shew by the figure; yet they seem to have done it for this reason, that unless it were in geometry, (in which only there is place for such figures) there was no ratiocination certain, and ending in science, their doctrines concerning all other things being nothing but controversy and clamour; which, nevertheless, happened, not because the truth to which they pretended could not be made evident without figures, but because they wanted true principles, from which they might derive their ratiocination; and, therefore, there is no reason but that if true definitions were premised in all sorts of doctrines, the demonstrations also would be true.
Properties of a demonstration, and order of things to be demonstrated.
17. It. is proper to methodical demonstration,
First, that there be a true succession of one reason to another, according to the rules of syllogizing delivered above.
Secondly, that the premises of all syllogisms be demonstrated from the first definitions.
Thirdly, that after definitions, he that teaches or demonstrates any thing, proceed in the same method by which he found it out; namely, that in the first place those things be demonstrated, which immediately succeed to universal definitions (in which is contained that part of philosophy which is called philosophia prima). Next, those things which may be demonstrated by simple motion (in which geometry consists). After geometry, such things as may be taught or shewed by manifest action, that is, by thrusting from, or pulling towards. And after these, the motion or mutation of the invisible parts of things, and the doctrine of sense and imaginations, and of the internal passions, especially those of men, in which are comprehended the grounds of civil duties, or civil philosophy; which takes up the last place. And that this method ought to be kept in all sorts of philosophy, is evident from hence, that such things as I have said are to be taught last, cannot be demonstrated, till such as are propounded to be first treated of, be fully understood. Of which method no other example can be given, but that treatise of the elements of philosophy, which I shall begin in the next chapter, and continue to the end of the work.