and that unity and the multitude should not be an element, for each monad is not from multitude and an unit. Further, the person who puts forward this assertion does nothing else than make another number, for multitude is a number of indivisible things.

7. This con

question as to number being finite or infimite;

and as to a

point, &c., what

Moreover, also, it is worthy of inquiry, in nected with the respect of those who make assertions in this way, whether number may be infinite or finite?1 for, as it appears, the multitude was also finite out of which and unity finite monads were produced, and multitude itself is different from infinite multitude. What sort of multitude, then, and what sort of an element, is unity? And in like manner might oL inquire, also, respecting a point and the element, from which does it subsist they make magnitudes; for there is not merely, at least, one actual point. Therefore, at any rate, one might ask the question from what each of the rest of the points will ensue? for, undoubtedly, it is not from a certain interval, at least, and an actual point. But, assuredly, neither is it admissible that indivisible portions constitute the portions of an interval, as they do of the multitude from which the monads consist, for number is composed of things that are indivisible; but this is not the case with magnitudes.

from?

drawn.

8. Conclusion Now, all these statements, as well as others of this kind, render it evident that it is an impossibility for number and for magnitudes to possess a separable subsistence.

9. In the discordancy of their opinions these specula

tors tacitly

allow the falsebood of their theories.

Moreover, the discordancy of the original framers of this Theory respecting numbers is an indication that these things, not being true, are fraught with sources of confusion unto them. For some of this school constituting mathematical natures merely in addition to those that are cognisant by the senses, when they came to perceive the difficulty and fiction attendant upon forms, have withdrawn their assent from the ideal or formal number,2 and have introduced mathematical number in its stead; but others wishing to make forms to exist at the same time with the

1 This point has been discussed in chapter viii.

2 Aristotle means the Pythagoreans.

numbers,1 but not discerning in what manner-on the supposition of one's admitting these as first principles-mathematical number will subsist independent of that which is ideal, have constituted ideal and mathematical number as the same in definition; since, in point of fact, at least, mathematical number has been done away with in this hypothesis: for they introduce peculiar theories of their own, and such as are not consistent with mathematical science.

10. Plato alone

view of the subject

The philosopher, however, who first sought to establish the existence of both forms and num- took a true bers, in obedience to the dictates of reason assigns a separate subsistence to forms and mathematical entities. Wherefore, it happens that all of this sect express themselves correctly in a certain respect, no doubt, yet not entirely with correctness. And themselves, likewise, acknowledge so much, as being persons who do not make the same statements at all times, but such as are contrary with one another.

But

not

11. The incon

sistency of the Pythagorics a falsehood of

result of the

And a cause of this is the following, that their suppositions and first principles are false. it would be difficult from things that are properly disposed in regard of truth and false hood to frame an hypothesis with correctness, according to Epicharmus; for in this case, as soon as the assertion is made, immediately also is apparent that which is not properly disposed in the before-mentioned respect.

their principles.

Regarding numbers, however, let thus much 12. Examina suffice of the questions that have been started, tion concerning and of the definitions and distinctions that have numbersended. been framed. For a person who has been brought to a state of acquiescence in a theory would still the more be induced to yield assent from the force of more numerous arguments; but nothing further will prevail towards inducing persuasion in the case of one who has not been prevailed upon to yield his assent already.

With respect, however, to first principles, and first causes, and elements, whatever assertions

1 Such as Speusippus and Xenocrates.

13. The theories ralists irrele

of the Natu

This is Plato, who recognised the existence of both forms and numbers, but contended for their subsistence distinctively, whereas tae Xenocratic dogma was to identify them.

3 Some make chapter xii to commence here.

vant to ontology; not so those of the Supranaturalists.

those persons put forward, who are engaged in framing1 their distinctions in regard of a substance merely cognisant by the senses, some of these, indeed, have been declared in our Treatise on Physics; but the remainder of them are omitted, seeing that they do not belong unto the plan of inquiry proposed to be pursued in our present Work. But whatever assertions are made by those who affirm that there exist different substances independent of those that fall under the notice of our senses, this is a subject for investigation consecutive to those statements that have been already made upon this point.

14. Amongst the Supranaturalists

some put for

ward numbers,

and some ideas,

of things.

Since, therefore, certain persons affirm that there are such like ideas and numbers, and that the elements of these are elements and first

principles of entities, with respect to these we as the original must inquire what it is they say, and how they say it. Those philosophers, then, who are for constituting as such existences numbers2 only, and such as are mathematical numbers, are to form subjects for examination afterwards.

15. Two fundamental mistakes of the Idealists, and the source of them.

Of those, however, who affirm the existence of the ideas, one should at the same time be able to perceive both the manner of their existence, and the matter of doubt that is prevalent regarding them; for also do they constitute ideas as existing simultaneously with universal substances, and, again, they view them as involving a separate subsistence even from singulars. But that these statements are not possible has been previously made a matter of doubt. A cause, however, of their connecting these substances into one and the same species-I mean, with those persons who call ideas universals -is because they are not accustomed to constitute them as the same substances with sensibles.

16. The Idealists cannot claim Socrates

as a patron of their system.

Some singulars, indeed, therefore, that are involved in objects that fall under the notice of our senses they considered to be in a state of flux, and not one of them to remain in a condition of permanence; but that the universal subsists both beside these and is something that is different from them. But, as 1 Aristotle has likewise examined these points in book I., and in Physics, book L 2 Vide book Xlll.

we have declared in the foregoing statements, Socrates communicated an inpulse, it is true, to this inquiry, by reason of definitions, yet he did not really abstract them, at least, from singulars; and, in thus not assigning them a separate subsistence, he formed his conceptions correctly.

And one could make this assertion evident 17. Socrates in from the actual occurrence of facts; for without this theory is universals, of course, it is not possible to attain matters of fact. supported by unto scientific knowledge: but the abstraction of

them from singulars is a cause of the difficulties that ensue in regard of ideas.

sals.

But some, as if it were necessary that if there are certain substances beside those that are cog- of this theory 18. The origin nisant by sense and are in a state of flux, they about univershould involve a separate subsistence-some, I say, were not in possession of other natures, but brought forward those that are denominated universals; so that it happens that both universals and singulars are nearly the same natures. This, to be sure, then, would itself amount to a certain essential difficulty in those statements that have been put forward above.

CHAPTER X.

WHAT it is, however, that is attended with 1. Repetition

doubt, both unto those who affirm the existence of statements. of ideas, and those who deny their existence, has, ideas. in regard of likewise, been observed previously,1 in the doubts enumerated at the beginning of this Treatise; let us, however at present, make a repetition of the statements made there. For if, indeed, one will not admit that substances involve a separate subsistence, and that the singulars of entities subsist in that manner as they are declared to do, such a view of things will overturn substance, as we are disposed to allow; yet, should one assume that there are substances possessing a separate subsistence, how will he establish the elements and the first principles of them?

elements of

For, supposing them to subsist as a singular, and 2. Results of not as an universal, entities of this kind will be supposing the 1 Vide book II. chap. ii.

separable sub

list as a sin

gular and not as universals; -illustrated

in a word.

as numerous as elements, and the elements will stances to sub- not be things capable of being made objects of scientific knowledge. For let the syllables in a word be granted to be as substances, and let the by the syllables elements of them be the elements of substances, in such a case as this it is, therefore, necessary that BA be one, and that each of the syllables should be one, if not, in fact, universally and the same in species, yet each must be one in number, and this certain particular thing, and not equivocal; and, further, they regard each one as the very thing itself. If syllables, however, be thus, so also will those things be of which syllables are composed. There will not, accordingly, be more than one letter A, nor will of the rest of the elements be more than one according any to the very same mode of reasoning, in accordance with which neither is there any of the other syllables that is the same; but there is one in one word, and another in another. But, certainly, if this be the case, there will not exist any different entities beside the elements; but entities will constitute elements merely. And, further, neither will the elements be objects of scientific knowledge, for they are not universals; but scientific knowledge is conversant about universals as objects of investigation.

3. Confirmed

of demonstra

tion and definition.

Now this is evident both from demonstrations1 from the nature and definitions;2 for a syllogism is not completed because this particular triangle has angles equal to two right angles, unless every triangle has angles equal to two right angles; nor because this man is au animal, unless every man is an animal.

4. If, then, first principles be universal, would not nonsubstance be prior to substance?

But, undoubtedly, if first principles are universal, or, also, if substances that are compounds of these are universal, non-substance in such a case will be a thing that is antecedent to substance; for, what is universal does not constitute substance: whereas the element and the first principle are universal. The element, however, and the first principle are things that are antecedent to those to which a first principle and an element belong. And, there

1 As might be seen in the course of argument which Aristotle pur gues in the Posterior Analytics.

2 As is done in book VI. of this Treatise.