Imatges de pÓgina
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Every propofition is, by geometricians, demonftrated either from axioms, that is, felf-evident truths; or fuch as have been elsewhere demonftrated from those which are felf-evident.

In like manner, whatever we propofe to demonftrate, the last appeal lies to felf-evident truths; in moral fubjects, to consciousnefs, or internal feelings; and in matters of revelation, to the plain sense of scripture and it is very expedient and adviseable, in discourses upon important fubjects of any kind, after the manner of geometricians, to premise these self-evident truths, beyond which no appeal can be admitted.

Moreover, left there fhould be any disagreement or dispute about the use of the words employed in the argument, it is, likewise, convenient that, after their manner, these axioms be preceded by definitions explaining the fenfe in which all the important words which represent complex ideas are used. When, in this manner, it is determined in what fense words are to be used, and what are the allowed uncontroverted principles we are to go upon, they may be applied with great ease and certainty in the remainder of the discourse; and the demonstration in which they are introduced, will be freed from that confufion and embarraffment which would otherwise attend it.

Befides, this method is, in a manner, the very touchstone of truth; and therefore, if our views really be to promote the intereft of truth (and fooner would I teach the art of poisoning than that of sophistry) this method hath another great advantage to recommend it. For if these definitions and axioms be laid down with due accuracy and circumfpection, they not only introduce the easiest, the moft natural, and cogent method of demonftrating any propofition, but lead to an easy method of examining the ftrength or weakness of the ensuing arguments. If

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the argument in fuch a methodical discourse be not conclufive, it contains within itself the principles of its own confutation. Such a discourse must be evidently inconfiftent with itself. On the other hand, if the definitions and axioms be admitted, the propofitions that are demonstrated from them, by the simple rules of reasoning, must be next to self-evident, and carry the strongest possible conviction along with them..

I am not, in these and the following obfervations, pleading for the geometrical TERMS, axiom and definition, or for the very exact and precife method in which geometricians place them. It is not the name, but the thing that I recommend; and only fo far as reafon directs to fimilar methods in fimilar cafes. A regard to perfpicuity would direct us (if we would be understood) to explain. distinctly the meaning of every word we ufe, that is of the least doubtful fignification, and to introduce the definitions, if not formally, at the entrance of a difcourfe, yet as foon as they become neceffary. It is manifeftly convenient likewife, upon feveral occafions, to refer exprefsly to maxims which are univerfally allowed or felf-evident, in order to fhow diftinctly upon what foundation an argument refts. The more diftinct we keep our own propofitions, or those which, in any discourse, we profefs to maintain, from thofe, by the help or medium of which, we prove them, the better. We can much more eafily examine any sentiments when we fee in what place to begin, and are fhown their mutual connexion, and the dependance that one part. hath upon another.

LECTURE

LECTURE VIII.

Of the feveral parts of a proper DEMONSTRATION.

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FTER these useful preliminaries, viz. ascertaining the use of terms, and premising what is univerfally known, or taken for granted, with respect to a subject, the geometrician proceeds to his propofition, in which he lays down, in the plainest terms, what he hath farther to advance. This either conftitutes a fingle propofition, or is refolvable into feveral heads, each of which are distinct propofitions, and must be demonstrated separately. Moreover, the principal propofition is fometimes preceded by one, or several others, which are called lemmas, and are defigned to prepare the way for the principal propofition, by proving the truth of fuch other propofitions as may be made use of to demonstrate it.

In like manner, if, when we have taken a view of the whole of a fubject, in all its extent, and have confidered every argument which we intend to bring in proof of it, we suspect that any of the intermediate propofitions, upon which the demonftration principally depends, may themselves want proof, or illustration, it may be extremely convenient to dispatch it in the introduction, previous to our naming the principal propofition; because it may prevent its occafioning any interruption in the course

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of the demonftration. Such doubtful positions must otherwise be propofed by way of answering objections, after the demonstration, which may not always be quite convenient; because the difficulty have occurred to the mind of the hearer, or reader, from the firft; and his keeping it in view through the whole of the demonftration, may have prevented the arguments from being heard with that attention, and freedom from prejudice, with which they would have been heard, if that objection had been obviated by way of lemma, in the introduction. The geometrician wifely anticipates all objection.

In some cases, indeed, it may be impoffible to anticipate all objections; as they may be of fuch a nature as that they could not be understood till the demonstration had been heard. In that cafe the objections not only may come after the demonstration (as of neceffity they muft, if they be mentioned at all) but also may do fo without any inconvenience. Because if the objection could not be understood before the demonftration, it could not have occurred to the hearer or reader before, fo as to lay any bias upon his mind in the course of it.

Objections being thus, as far as poffible, anticipated, and the truth of every intermediate propofition that we shall have occafion for, proved, the way is properly cleared for the principal propofition, which must be proposed without any ornament, in the most intelligible terms. If the propofition be complex, the whole extent of it must be shown in the moft commodious divifion of it into its proper parts: alfo the order in which each part will be difcuffed must be pointed out diftinctly, that the whole procefs of the demonftration may lie with the greatest clearness before the minds of those to whom it is addreffed; and that, in the progress of the discourse, they may perceive the connexion of all the parts,

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parts, and may fee all along what progress the speaker or writer hath made in his argument.

In cafes relating to matters of fact, it may require a long and circumftantial narration before the point to be proved can be understood. Whatever narration, therefore, is requifite to fet as question in difpute in a clear light, belongs to this part of a dif courfe, and is properly referred to the propofition.

The geometrician, when he hath laid down his propofition, proceeds, by a series of fteps which terminate in a single proof, to show the agreement or coincidence of the terms of it: and as one demonstration, in fubjects that will admit of it, is decifive, a multiplicity and redundancy of proofs is feldom affected by mathematicians. But in this the moralift and divine must con tent themselves with following them at a great and very humble distance. As the fubjects they treat of are not always capable of ftrict demonftration, they are obliged to have recourse to a variety of arguments, each of which may add fomething to probability, (which in its own nature admits of degrees) till the united ftrength of them all be fufficient to determine the affent.,

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In this cafe, it is of fome confequence that attention be paid to the order of the proofs, fuppofing them to be of different natures, and different degrees of ftrength. Arguments of a similar nature, that is, drawn from fimilar confiderations, as from reafon or fcripture, obfervation or experience, &c. fhould be ranged together; because in that position they confirm, and throw light upon one another. And though arguments which have no weight ought by no means to be used at all, and one that hath but little weight had better be fpared, where there are a fufficient number of substantial and Atriking arguments, yet in some cases it may

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