[85] That subtraction ¹ is nothing they argue thus: If anything is subtracted from anything, either equal is subtracted from equal, or greater from less, or less from greater. But in none of these ways does subtraction take place, as we shall show; therefore subtraction is impossible.

That subtraction takes place in none of these ways is plain from what follows: What is subtracted from anything ought, before its subtraction, to be included in that from which it is subtracted.

[86] But the equal is not included in the equal—six, for instance, in six; for what includes must be greater than what is included, and that from which the subtraction is made than what is subtracted, in order that there may be some remainder after the subtraction; for it is this which is held to distinguish subtraction from complete removal. Nor is the greater included in the less—six, for instance, in five; for that is irrational.

[87] And for this reason, neither is the less included in the greater. For if five is included in six, as less in greater, four will be included in five, three in four, two in three, and one in two. Therefore six will contain five, four, three, two, and one, which when put together form the number fifteen, and this we conclude is included in six, if it be granted that the less is included in the greater. So likewise in the fifteen which is included in the six there is included the number thirty-five, ² and so on, step by step, to infinity. But it is absurd to say that infinite numbers are included in the number six; and so it is also absurd to say that less is included in greater.

[88] If, then, what is subtracted from a thing must be included in that from which it is to be subtracted, and neither equal is included in equal, nor greater in less, nor less in greater, then nothing is subtracted from anything.

Again, if anything is subtracted from anything, it is either a whole subtracted from a whole, or a part from a part, or a whole from a part, or a part from a whole.

[89] But to say that a whole is subtracted from either a whole or a part is plainly nonsense. It remains, then, to say that a part is subtracted either from a whole or from a part; which is absurd. Thus for example—basing our argument on numbers for the sake of clearness—let us take ten and suppose that from it one is subtracted. This one, then, cannot be subtracted either from the whole ten or from the remaining part of the ten, as I shall show; therefore it is not subtracted at all. ³

[90] For if the one is subtracted from the whole ten, since the ten is neither something other than the ten ones nor one of the ones, but the aggregate of the ones, the one ought to be subtracted from each of the ones in order to be subtracted from the whole ten. Now from a one, above all, nothing can be subtracted; for the ones are indivisible, and on this account the one will not be subtracted from the ten in this way.

[91] And even were we to grant that the one is subtracted from each of the ones, the one will contain ten parts, and as containing ten parts it will beaten. And further, since ten other parts remain, after the subtraction of the ten parts of the so-called one, the ten will be twenty. But it is absurd to say that the one is ten and the ten twenty, and to divide what, according to them, is indivisible. ⁴ Wherefore it is absurd to say that the one is subtracted from the whole ten.

[92] Neither is the one subtracted from the remaining nine; for that from which anything is subtracted does not remain entire, but the nine does remain entire after the subtraction of that one. Besides, since the nine is nothing more than the nine ones, if it should be said that the one is subtracted from the whole nine, the sum subtracted will be nine, or if from a part of it, then in case it be eight the same absurd results will follow, while if the subtraction is made from the last one, they will be affirming the divisibility of the one, which is absurd.

[93] So then, neither from the nine is the one subtracted. But if it is neither subtracted from the whole ten nor from a part of it, no part can be subtracted from either a whole or a part. If, then, nothing is subtracted either as whole from whole or as part from whole, nor as whole from part or as part from part, then nothing is subtracted from anything.

[94] Moreover, addition is regarded by them ⁵ as one of the impossibles. For, they say, that which is added is added either to itself or to what pre-exists or to the compound of both; but none of these alternatives is sound; therefore nothing is added to anything. Suppose, for instance, a measure of four cups, ⁶ and add to this a cup. To what, I ask, is it added? for it cannot be added to itself, since what is added must be other than that whereto it is added, but nothing is other than itself.

[95] Neither is it added to the compound of the four cups and the one cup; for how could anything be added to what does not yet exist? Besides, if the added cup is blended with the four cups and the one cup, six cups will be the measure resulting from the four cups and the one cup and the added cup.

[96] And if the, cup is added to the four cups alone, since that which is extended over anything is equal to that over which it extends, the cup which extends over the measure of four cups will double the four cups so that the whole measure becomes eight cups—a result contrary to experience. If, then, what is said to be added is neither added to itself nor to what pre-exists nor to the compound of these, and besides these there are no other alternatives, then there is no addition of anything to anything.