Therefore, I ask, first, whether a point is an absolute thing really distinct from quantity. Which is answered affirmatively; for the terminus of every single thing is distinguished from that of which it is a terminus. But a point is a terminus of a line; therefore, it is really distinguished from a line, and consequently it is for the same reason really distinguished from any quantity.
On the contrary, every positive thing is substance or quantity or quality or relation, and so of the other categories. But a point is not a substance distinct from quantity, nor a quality distinct from quantity, nor a relation distinct from quantity, and so of other respective genera; therefore, a point is not a thing other than quantity.
Before answering this question and all following concerning this matter, I make in advance one statement; namely, that whatever I shall say under whatever form of words, which can in any mode be regarded as contrary to whatever has been said in Sacred Scripture or in the writings of the Saints, or contrary to the determination and the doctrine of the Roman Church, or contrary to the opinion of the Doctors approved by the church, I shall not say by affirming but by quoting precisely in the person of those who also hold the opinion to be discussed; whether that opinion be true or false, or Catholic, or heretical, or erroneous. Whence, if I shall use such words as “I say,” “It must be said,” and the like; I desire those words to be understood not as used by me personally but as used by those who hold such opinions.
With this understood, I shall proceed thus with respect to the question proposed. First, I shall show that a point is not some positive and absolute thing really distinct from any quantity and especially from a line. In the second place, I shall posit some reasons and passages to prove that a point is another thing. In the third place, I shall reply to these.
First, therefore, I prove that a point is not a thing other than a line, or any quantity; and in the first place, through philosophical reasons; in the second place, through passages of the philosophers; in the third place, through theological reasons.
According to the first way; I, first, argue ostensively thus: if a point be a thing other than quantity, it is either substance or accident; but a point is not substance distinct from quantity nor accident distinct from quantity; therefore, a point is not a thing other than quantity.
The major premise is known; the minor premise I prove. And, first, I prove the first part; namely, that a point is not a substance distinct from quantity; for every substance is either matter or form or a composite or substance abstracted from matter. But it is manifest that a point is not matter or form or a composite; nor is it form abstracted from matter, if a point be indivisible and in no way a quantum, as is posited by the opponents; therefore, et cetera.
I prove the second part thus; namely, that a point may not be an indivisible accident distinct from quantity. In the first place thus: every absolute accident is placed in any subject adequate to it, so that that accident is either a whole in a whole subject and in each part, or a whole in a whole and a part in a part or is an indivisible accident having an indivisible subject, as a spiritual accident is in the rational soul as if in its primary and adequate subject. But a point in none of these three modes has or can have any primary and adequate subject; therefore, the major premise of that syllogism is obvious through induction as regards any absolute accident.
The first part of the minor premise is obvious; namely, that a point does not have any adequate subject so that there may be a whole in a whole and in each part. For I take a point terminating the line A, and let that be called B; and I take another corresponding point, and let it be called C; then I ask whether B is in A, as if in its adequate subject or in the subject of itself; and it follows in either way, that C will be in the same adequate subject for the same reason.
Then I argue thus: whatever things are in the same adequate subject, they do not differ either in situation or in place; but B and C are distinguished or separate in place and in situation; therefore, they are not in the same adequate subject. If it be said that B is not in the whole line A, nor in the subject of the same line A, but in a part of the same line or in part of the subject (of the line); on the contrary, if B be not in the whole line as if in an adequate subject nor in the whole subject of the same line; then it is in a part as if in an adequate subject. Let that part be D; then I argue, B is in D as if in an adequate subject; therefore, for the same reason another point corresponding to the same B, at which D is terminated out of another part, will be in the same D as if in an adequate subject, and let that point be called E. From which it follows that B and E are in the same adequate subject; and consequently they are not separate in place and in situation, which is most manifestly false. So, therefore, there can not be any point in any divisible quantum, and as if in its adequate subject, so that a whole may be in a whole and a whole in each part; not only because two points of that divisible would not then be separate in situation, which is false; but also because a point the same numerically would then be in two situations, which is false.