ABSTRACT
WE have in Part delineated the Nature of Geometrical Reason by explaining and vindicating its Definition assigned in the Elements. Which might have been otherwise declared by saying more generally and confusedly, but perhaps more agreeable to a common Capacity, that Reason is the determinate Way or Manner whereby any one Quantity contains, or is contained in, another, which is the same, or very like to that Way or Manner according to which one Quantity is said to be so many Fold of another, or to contain another so often, or to be such a Part or so many Parts of another, or to be contained so often in another: which Manner is easily comprehended, when the Quantities compared can be expressed in Numbers; as the Manner, whereby a Pound Troy contains an Ounce, signifies no more than that the former contains the latter twelve Times, or is to it as 12 to 1; whence a Pound is said to obtain a Duodecuple Reason to an Ounce. The Manner whereby a Foot in Length is contained in a Geometrical Pace is that the former is the fifth Part of the latter, or is to it as 1 to 5, which is signified by saying, that the Reason of a Foot to a Pace is subquintuple: Which Manner is expressed by the abstract Word Quintuplicity or Quintuplity. But when the Quantities compared, as to their absolute Quantity, are such as cannot be perfectly expressed, at least they may be understood in Numbers; because that contains, or is contained in this, after some determinate Manner, otherwise than it contains, or is contained in, any other Thing unequal to this; therefore this Way or Manner of containing is very like that, whereby the Quantities denominated by Numbers respectively do contain, or are contained in, one another, and may always be nearly expressed in Numbers simply, and on the Part of the Thing. As ex. gr. when the Reason of the Periphery of a Circle to its Diameter (though not exactly and perfectly known) is represented by saying that the Periphery by Apposition is triple and sesquiseptimal, i. e. contains the Diameter three times and one seventh Part of it nearly. So also the Reason of the Diameter to the Side of a Square, which the Nature of the Thing will not admit to be precisely expressed in Numbers may yet be represented approaching the Truth, by saying that the Diameter is to the Side as 1.4 to 1, or more nearly as 1.41 to 1, or yet again more nearly as 1.416 to 1, &c.
