Treatise IV. On the hunting of definition, which is the discovery of the middle of demonstration by the way of discovery. Chapter I. How one proceeds through division in the hunting of the definition of a subject.

TREATISE IV. On the hunting of definition, which is the discovery of the middle of demonstration by the way of discovery. CHAPTER I. How one proceeds through division in the hunting of the definition of a subject. But in what way one must hunt in that which belongs to what it is, things to be hunted, that is, things hunted out, or things to be received by the hunting of definers, we shall now say in this treatise. Therefore first in this one must receive that, of those things which belong to each thing as substantial predicates, some according to the commonness and scope of their predication are extended more widely than the commonness of the subject or species which must be defined, yet in such a way that that scope of predication is not extended outside the genus of that which must be defined. But I say by explaining that those things are said to be extended more widely, that is, to a greater commonness, which belong universally to some other thing to be defined and are universally predicated of it; but nevertheless they do not belong to that alone, but also to another. Thus by way of example one may say: there is something which belongs to every trinity, that is, ternary, and not to it alone, but also to another species of number which is not trinity or ternary. But this is twofold, because either it is outside the genus or in the same genus, which is number. If it is outside the genus, then it is as that which is, that is, being which transcends every genus, belongs to trinity or ternary and to non-trinity, that is, non-ternary; but it is outside the genus, because it belongs to that which is not number, as to whiteness, and therefore must be removed from the definition, because in definition only that which is in the same genus with the defined thing is posited. But odd, if it is taken as a difference of number, is such a predicate that it belongs to every trinity, and is more widely than trinity, because it does not belong only to trinity or ternary, since five also is an odd number; therefore it is more widely than trinity. And indeed, as we said, it belongs to five itself, that is, to the quinary, which is not trinity, although it is in the same genus of number in which trinity or ternary is. For five is a number, and no thing which is to be posited outside number is odd; and therefore the quinary is in the same genus with the ternary. For such things dividing the genus, namely the genus of that which must be defined, are to be received, and the things dividing the genus by division and subdivision, until by subdividing so many are received that each of the things received, taken according to itself, is more widely in act or potency than that which must be defined, but the whole gathered from all the dividing things is equally extensive with the thing to be defined and not more widely than that very thing which must be defined. For it is necessary that this expression gathered thus from all the dividing things ordered to that be the substance, that is, the substantial definition, of the thing to be defined, which is a species as the subject of demonstration; just as it belongs to trinity as its definition, that it is an odd number, first in both ways. For when number which is odd is divided, it also is odd; and therefore it is an odd number, and is first in both ways, being composed from each number, that is, from unity, which is first in number insofar as it is an aggregation of units, and from twoness, which is first and the principle of multitude insofar as number is multitude, and it does not come to be from the multiplication of number into number. And although these things accepted together, namely odd number first in both ways, are a definition convertible with ternary, nevertheless number is more widely, and odd is more widely, and being first from each is more widely. And concerning number and odd this is clear per se. But that which is to be first in both ways is also more widely, because it also belongs to the binary, which is not from the multiplication of some number into some number, but from the first aggregation of units, which makes the first number. But the whole together belongs only to ternary or trinity, because the binary is not odd; and no number between the binary is first in both ways except the ternary. And therefore this definition gathers all things which are substantial for trinity and separates it from all others. Therefore it is clear that it is the definition of it. But thus it also belongs to trinity that it is a number which is not measured by numbers or by a number, and just as it belongs to the same not to be composed from numbers, but from number and unity, which is not number. Thus therefore one must proceed: number is either odd or even; and it is agreed that it is not even, but odd; therefore ternary is an odd number. And number is either first in both ways or not first in both ways. And if it is number, it is either measured by number or not; and it is agreed that it is not a number measured by number; therefore trinity is an odd number, first in both ways, not measured by number. Moreover, if it is number, it is either composed from numbers or not; and it is agreed that it is not composed from numbers. Therefore trinity is an odd number, first in both ways, not measured by numbers, nor composed from numbers. And let it be posited that the division of number stands there. Therefore this expression gathered from all these things which are predicated universally and substantially in this division of ternary is, through definition, substantial trinity or ternary, which is the first odd number; and thus first it belongs to it that it is an odd number. For each of these, that is, each one, belongs to all odd numbers, because every odd number is a number, but it is not converted. But the last thing, which is to be first in both ways, and not to be measured by number, and not to be composed from numbers, belongs indeed to every duality, and so does any one of them; but all these things gathered together belong to no one except the ternary. And thus the definition of ternary is an odd number, first in both ways, uncomposed, and measured by no number. But since it has now been shown by us in what was had before that those things which are necessary are those which are predicated of something in that which is what it is, and that those things which are so predicated of something are universal, higher, and more common predicates which are predicated of every thing always, therefore such predicates enter into definitions, which are in the first mode of saying per se, in which the predicate is in the account of the subject. And such predicates concerning trinity, that is, ternary, and also in another number concerning which such predicates are predicated in that which is what it is, so that they are among the definitions of those subjects concerning which they are so predicated; and therefore, as has been said, when they have thus been received through definition and gathered together, they will here be gathered together through the definition of trinity, that is, ternary exists according to what it is and by definition. But that substance, that is, the substantial definition of each thing, is from these, namely from substantial things thus received through definition, is manifest. For it is necessary and follows necessarily that unless what has been gathered in the stated way were the being and definition of trinity, that is, ternary, since it is predicated in what, and such a predicate in what and not in what sort is either definition or genus. Therefore if such a predicate is not definition, it follows necessarily that it is genus. But if it is conceded that it is genus, then it follows that it is predicated without conversion of that of which it is the genus, because genus is more widely than to belong only to trinity; for a genus belongs to several species. For it was conceded and proved in what was had before that genus is a predicate of this sort which is more widely than the species of which it is predicated, according to the scope and potency of its commonness. But such a predicate is not a definition, as was said, when gathered. Therefore if it is not genus, it follows therefore that it is a predicate as definition; and if it will belong to no other things than to atoms, that is, singulars, and to things indivisible according to the commonness of trinity, as to this ternary and to that, and so concerning other ternaries, then it follows that this thing thus gathered is the being, that is, the essential being definition, of trinity or ternary; for it belongs to trinity alone and to every atom of trinity, and is a predicate in what it is. For let this be supposed from what has been said and be held as a supposition, that this is the substance and substantial definition of each thing, namely what is the ultimate predicamental item or predicate which is not further divisible, and therefore convertible with the defined most-specific species. For this reason, as was said about ternary and its definition, so it is in all other most-specific species, such that for any most-specific species that will be its definition which has thus finally been gathered from substantial differences and genus in all things which are thus demonstrated, as most-specific species and things to be defined are proposed. <marginal-label>Doubt.</marginal-label> But one must attend that it has been said that the individual defining things are more widely than the defined species, and the whole gathered is equally extensive, which is against what is said in the seventh book of the First Philosophy ¹, where it is shown that the ultimate difference is convertible. But although this has often been solved by us, nevertheless here it can be said that if division is led down to atoms which are further indivisible according to common form, the ultimate predicate, which is difference, must be convertible with the species, because otherwise it would be further divisible. But because ultimate differences insofar as they are divisive are unnamed, while the more common are named, therefore it is said that among named differences any one is more common than the most-specific defined species. <marginal-label>Solution of certain persons.</marginal-label> Yet certain people say, and quite reasonably, that division can be in two ways, namely through ultimate things or through those things which are before the ultimate. And if it is through ultimate things, then what is said in the seventh book of the First Philosophy is plain, because those are convertible. But if it is not through ultimate things, either they are received by one, that is, by a division of one, or by several divisions. If by one division, again it is a division through ultimate things and of the ultimate genus which is divisible, and is proximate to the most-specific species; and thus again the ultimate things are convertible. But if they are received by several divisions, either those divisions are received ordered from the first to the last, or as coaequal, so that one thing is divided by several coaequal things, of which one is not under another. The first example is substance, which is divided by corporeal and incorporeal; and afterward corporeal by animate and inanimate; and thus the division runs down. And then such divisive differences, received according to individual ones, are always more widely than the most-specific species which must be defined; and concerning such things what has been said here is true, because a difference has the name of difference from that which is divisive and not from that which is constitutive. But if several divisions are received not ordered, but among coaequal things, then what has been said is plainly true, namely that the individual things are more widely and the whole is equally extensive. The example of this is if number is divided, as was said a little before, into number first in both ways and not first in both ways; and again number is divided into what is composed from numbers and what is not composed from numbers; and again it is divided into number measured by number and into what is not measured. All the divisions are equal to that which is divided and are not ordered by succession, so that that which divides above is subdivided and thus the division runs down to the ultimate things; and in such a division the accepted differences are more widely. But the whole gathered is made equally extensive because of the denial of the other and the contradiction to one another in the individual divisions, as rational excludes irrational things and mortal excludes immortal things. And in this mode what has been said is plain: the parts are more widely, but the whole is equally extensive.