------- CHAPTER XIII. OF ANALOGISM, OR THE SAME PROPORTION. 1, 2, 3, 4. The nature and definition of proportion, arithmetical and geometrical.—5. The definition, and some properties of the same arithmetical proportion.—6, 7. The definition and transmutations of analogism, or the same geometrical proportion.—8, 9.. The definitions of hyperlogism and hypologism, that is, of greater and less proportion, and their transmutations.—10, 11, 12. Comparison of analogical quantities, according to magnitude.—13, 14, 15. Composition of proportions.—16, 17, 18, 19, 20, 21, 22, 23, 24, 25. The definition and properties of continual proportion.—26, 27, 28, 29. Comparison of arithmetical and geometrical proportions. [Note, that in this chapter the sign + signifies that the quantities betwixt which it is put, are added together; and this sign - the remainder after the latter quantity is taken out of the former. So that A + B is equal to both A and B together; and where you see A - B, there A is the whole, B the part taken out of it, and A - B the remainder. Also, two letters, set together without any sign, signify, unless they belong to a figure, that one of the quantities is multiplied by the other; as A B signifies the product of A multiplied by B.] [Sidenote: The nature and definition of proportion, arithmetical & geometrical.] 1. Great and little are not intelligible, but by comparison. Now that, to which they are compared, is something exposed; that is, some magnitude either perceived by sense, or so defined by words, that it may be comprehended by the mind. Also that, to which any magnitude is compared, is either greater or less, or equal to it. And therefore proportion (which, as I have shewn, is the estimation or comprehension of magnitudes by comparison,) is threefold, namely, proportion of _equality_, that is, of equal to equal; or of _excess_, which is of the greater to the less; or of _defect_, which is the proportion of the less to the greater. Again, every one of these proportions is two-fold; for if it be asked concerning any magnitude given, how great it is, the answer may be made by comparing it two ways; first, by saying it is greater or less than another magnitude, by so much; as seven is less than ten, by three unities; and this is called _arithmetical proportion_. Secondly, by saying it is greater or less than another magnitude, by such a part or parts thereof; as seven is less than ten, by three tenth parts of the same ten. And though this proportion be not always explicable by number, yet it is a determinate proportion, and of a different kind from the former, and called _geometrical proportion_, and most commonly _proportion simply_.