The English works of Thomas Hobbes of Malmesbury, Volume 01 (of 11)

by Thomas Hobbes

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.

2. Proportion, whether it be arithmetical or geometrical, cannot be exposed but in two magnitudes, (of which the former is commonly called the antecedent, and the latter the consequent of the proportion) as I have shewn in the [8th article] of the preceding chapter. And, therefore, if two proportions be to be compared, there must be four magnitudes exposed, namely, two antecedents and two consequents; for though it happen sometimes that the consequent of the former proportion be the same with the antecedent of the latter, yet in that double comparison it must of necessity be twice numbered; so that there will be always four terms.

3. Of two proportions, whether they be arithmetical or geometrical, when the magnitudes compared in both (which Euclid, in the fifth definition of his sixth book, calls the quantities of proportions,) are equal, then one of the proportions cannot be either greater or less than the other; for one equality is neither greater nor less than another equality. But of two proportions of inequality, whether they be proportions of excess or of defect, one of them may be either greater or less than the other, or they may both be equal; for though there be propounded two magnitudes that are unequal to one another, yet there may be other two more, unequal, and other two equally unequal, and other two less unequal than the two which were propounded. And from hence it may be understood, that the proportions of excess and defect are quantity, being capable of more and less; but the proportion of equality is not quantity, because not capable neither of more, nor of less. And therefore proportions of inequality may be added together, or subtracted from one another, or be multiplied or divided by one another, or by number; but proportions of equality not so.

4. Two equal proportions are commonly called the same proportion; and, it is said, that the proportion of the first antecedent to the first consequent is the same with that of the second antecedent to the second consequent. And when four magnitudes are thus to one another in geometrical proportion, they are called proportionals; and by some, more briefly, analogism. And greater proportion is the proportion of a greater antecedent to the same consequent, or of the same antecedent to a less consequent; and when the proportion of the first antecedent to the first consequent is greater than that of the second antecedent to the second consequent, the four magnitudes, which are so to one another, may be called hyperlogism.

Less proportion is the proportion of a less antecedent to the same consequent, or of the same antecedent to a greater consequent; and when the proportion of the first antecedent to the first consequent is less than that of the second to the second, the four magnitudes may be called hypologism.

The definition and some properties of the same arithmetical proportion.

5. One arithmetical proportion is the same with another arithmetical proportion, when one of the antecedents exceeds its consequent, or is exceeded by it, as much as the other antecedent exceeds its consequent, or is exceeded by it. And therefore, in four magnitudes, arithmetically proportional, the sum of the extremes is equal to the sum of the means. For if A. B :: C. D be arithmetically proportional, and the difference on both sides be the same excess, or the same defect, E, then B + C (if A be greater than B) will be equal to A - E + C; and A + D will be equal to A + C - E; but A - E + C and A + C - E are equal. Or if A be less than B, then B + C will be equal to A + E + C; and A + D will be equal to A + C + E; but A + E + C and A + C + E are equal.

Also, if there be never so many magnitudes, arithmetically proportional, the sum of them all will be equal to the product of half the number of the terms multiplied by the sum of the extremes.