From a catena like the strongness or strength catena we can directly derive the strongness-moreness or 'strongerness' catena. But besides the derivations stronger or more strong, weaker or less strong and equally strong or equally weak we also know derivations like strengthening and weakening. There thus exists a strengthening catena as well. In other systems we will find an honoring catena, betterment catena, heating catena, and so on. All these comparative catenas are increase catenas as explicit triads or differentiation catenas as quasi-duads.

Differentiation stands to difference or otherness and different or other as strengthening stands to strongness and stronger, and as increase stands to moreness and more. Also differentiation- or increase-catenas can be subdivided into positivity-differentiation and neutrality-differentiation catenas (or any other type dependent on the aspectual value taken). And, analogously to the case of difference catenas, the increase catena of an explicit triad is a positivity-increase catena, and the increase catena of a bipolarity catena a neutrality-increase catena. These similarities are obvious. They hold for every increase or differentiation catena and the corresponding moreness- or difference-catena. Roughly speaking, the difference-catenary approach is primarily nontemporal, whereas the differentiation-catenary approach is primarily temporal, at least on the assumption that every comparatively catenal thing exists over a period of time.

Differentiation, increase or decrease can be active or passive. For example, active positivity-differentiation means making less or more positive(ly catenal) and passive positivity-differentiation becoming or growing less or more positive(ly catenal). Active neutrality-increase is making-more-neutral(ly catenal), and passive neutrality-increase, becoming- or growing-more-neutral(ly catenal). (Literally speaking, one cannot make something positive or neutral, or become positive or neutral. A primary predicate just is or is not positive or neutral, while it is a nonpredicative primary thing which can or cannot be made or become positively or neutrally catenal.) Each differentiation catena is, as it were, the common denominator of two catenas: an active (transitive) and a passive (intransitive) one. Both catenas are each other's isorelative but --as already explained in 2.3.3-- the passive variant consists of pseudo-attributes only.

The fact that a catena is a difference or differentiation catena, whether active or passive, says almost nothing about its position in the total derivation system, except that it is not a basic catena. Only if the original catena is basic, is the comparative catena a catena of the first level of derivation. In general, it is of the (n+1)-st level, if the original catena is of the n-th level. (Basic catenas are of the 0-level of derivation.)

Catenas of the same basis are interchangeable. For example, the longitude, latitude and altitude catenas are interchangeable in that the spatial co-ordinate which is regarded as longitude, could also be latitude or altitude instead, and vice versa. Catenas derived from two interchangeable catenas in the same way are interchangeable themselves. Interchangeable catenas are always equidimensional, but equidimensional catenas need not be interchangeable. The difference and differentiation catenas of the longitude catena, for instance, are equidimensional to it but not interchangeable with it (with the exception of the monovariant positivity-difference catena for which c=0, which is identical to it). Monocatenal and bicatenal difference catenas are also equidimensional, but not interchangeable either: the length of an object, for instance, or its shortness catenality, is something else than the distance between this object and another object, or its closeness catenality.

Unlike difference and differentiation catenas, which are equidimensional to the original catena from which they are derived, differential catenas have their own dimension. Consider, for example, velocity. If v represents an object's velocity, then v=ðs/ðt (with ðs as the distance traveled and ðt as the time spent); v is therefore s-differentiated-to-time, and its dimension is m/sec. On the basis of a terminological analogy between value collections and catena extensionalities, we may say that one can derive the motion catena by differentiating the longitude catena, or a catena interchangeable with it, to time. Thus we shall call the motion catena, the quasi-duad corresponding to velocity, "a time-differential catena". The term velocity is also used in the more general sense of rate of occurrence or action. In that case it corresponds to any time-differential catena, not just that in the derivation system of the longitude catena.

The differential catenas to time of the longitude, latitude and altitude catenas are mutually interchangeable. It may be said that each of the three time-differential catenas is a motion catena; it may also be said that the motion catena is the common denominator of these and similar catenas. This is no fundamental issue.

Starting from a differential catena which is derived from a basic catena, we can in turn derive a difference or other comparative catena from it. For example, in the physical system of the longitude catena, the slowness catena is a modulus-catena of such a differential catena. The dimension of this modulus-catena is the same as that of the differential catena, namely m/sec. But also when we now comparatively derive a new catena from the slowness catena, for example, the bivariant differentiation catena with extreme aspectual value, the dimension remains the same. Thus the dimension of this retardation catena is also m/sec. On the other hand, deceleration (and also acceleration) has the dimension m/sec² in physics. The catena to which this deceleration belongs, however, is not the differentiation catena of the slowness- or speed-catena, but the time-differential catena of the speed catena.

Just as a derivative and a first derivative are the same in mathematics, so a differential catena is the same as a first differential catena. (In the theory of catenas a derivative need not be a differential catena tho.) A second differential catena is, then, the differential catena of a differential catena with respect to the same form of catenality or quantity. The deceleration catena is no such second differential catena; it is the first differential catena of the modulus-catena of the first differential catena of the longitude catena or a catena with which it is interchangeable.

In traditional mathematics the differentials ðv or DELTAv and ðt or DELTAt are evaluated positive, if the new v- or t-value is closer to the positive extreme than the old one. The aspectual value is on this account supX or +INFIN. Now this view is typically that from the perspective of positivity-moreness, based on the positivity-increases of v and t. What underlies this conception is positivity-differentiation. The differential catena thus derived is a positivity-differential catena.

It is not possible to make a mathematical differentiation universally comparative, because the aspectual value will always determine the value of the derivative function. On the other hand, we are just as justified of looking at DELTAv and DELTAt from the perspective of neutrality-moreness, and to evaluate DELTAv and DELTAt positive if the new v or t is closer to zero than the old one, and negative if it is farther away from it. The catena thus derived is a neutrality-differential catena. If û₁ is the catena value of the original catena, and f⁺(û₁) the value of the positivity-differential catena, the value of the neutrality-differential catena is
û₂ = f⁰(û₁) = ( | f(û₁) | × û₁ × f⁺(û₁)) ÷ ( | û₁| × f(û₁)) for û₁¹0 and f(û₁)¹0 .
If û₁ ¹ 0 and f(û₁) = 0, then f⁰(û₁) = 0 regardless of the value of f⁺(û₁). (This is because we choose a point on the curve which is a point on the tangent midway between the lower and the higher triangular points.) If û₁ = 0 and either f(û₁)¹0 or f⁺(û₁)¹0, then f is indefinite; and if û₁ = 0 and both f(û₁)¹0 and f⁺(û₁)¹0, f⁰ is infinite (even if f⁺ is finite).

If A is the difference catena of B, then B may be called "a sum catena of A". Similarly, if C is the differential catena of D, D may be called "an integral catena of C". For example, the quasi-monad of energy can be conceived of as an integral catena of the physical force catena with respect to the length catena. Differential and integral catenas have to be distinguished from the related quotient and product catenas. The product catena of the physical force catena and the length catena would be the physical moment catena, while the integral catena is an energy catena. Nevertheless, differential and quotient catenas, and also integral and product catenas, have the same dimension, granted that the original catenas are the same.

As the kinds of catenical derivations are closely related to the kinds of mathematical operation it is worth our while to briefly consider the position of the most simple mathematical operations. The first pair of operations is, then, that of addition (a + b = c) and substraction (a - b = c). There is no repetition involved in these operations. In the theory of catenas it is the derivation of a comparative and equidimensional catena which is the analog of the mathematical sort of operation on this zero-level of reiteration. (All mathematical operations or functions may also be employed, however, to express logically contingent relationships between catena values, something which is not our concern here.)

The prototypical mathematical operation on the first level of reiteration is multiplication (a × b = c). This is in the first instance nothing else than a form of reiterative adding-up:
b × a = a + a + ... + a , or
b × a = SUM(i=1, b) a_i (a_i=a for every i).
Its correlative is division (a ÷ b=c). The derivations of nonequidimensional differential and integral catenas, and of quotient and product catenas, are the catenical analogs of mathematical operations on this level.

The prototypical mathematical operation on the second level of reiteration is involution or the raising of a quantity to power (a^b = c or a××b = c). This is in the first instance nothing else than a form of reiterative multiplication:
a ×× b = a × a × ... × a , or
a × b = PRODUCT(i=1,b) a_i (a_i=a for every i).
The correlative of this second-level operation is evolution, the extraction of a mathematical root ( b ROOT a=c ). In their original shape the numbers were what mathematicians traditionally call "natural".

It would be naive to take it for granted now that the number of sorts of mathematical operation is thus exhausted. For we can continue ad infinitum by repeating the operation on the previous level of reiteration. On the third level this would involve reiterative involution with its related forms of operation. But on this and higher levels ordinary, and also mathematical, language just lack the terminology to express ourselves, even if we would like to. Nevertheless, it is possible to develop a universal notational system for reiterative operations of all levels by means of novel mathematical symbols. Suffice it here to recognize that there are not only other catenas on the first level of reiteration besides differential catenas but also catenas on higher operational levels of reiteration. They are catenas of primary predicates for which there is even no name in scientific or technical noncatenical parlance.