Naturalists try to arrange the species, genera, and families in each class, on what is called the Natural System. But what is meant by this system? Some authors look at it merely as a scheme for arranging together those living objects which are most alike, and for separating those which are most unlike; or as an artificial means for enunciating, as briefly as possible, general propositions, that is, by one sentence to give the characters common, for instance, to all mammals, by another those common to all carnivora, by another those common to the dog-genus, and then by adding a single sentence, a full description is given of each kind of dog. The ingenuity and utility of this system are indisputable. But many naturalists think that something more is meant by the Natural System; they believe that it reveals the plan of the Creator; but unless it be specified whether order in time or space, or what else is meant by the plan of the Creator, it seems to me that nothing is thus added to our knowledge. Such expressions as that famous one of Linnaeus, and which we often meet with in a more or less concealed form, that the characters do not make the genus, but that the genus gives the characters, seem to imply that something more is included in our classification, than mere resemblance. I believe that something more is included; and that propinquity of descent, the only known cause of the similarity of organic beings, is the bond, hidden as it is by various degrees of modification, which is partially revealed to us by our classifications.
He argues this point throughout the chapter. We know that Darwin read Whewell; and we also know that some of Darwin's own conception of science was influenced by Whewell, although there appears to be no consensus as to the extent of that influence. But whether Darwin is influenced by Whewell here or not, he is inserting the theory of natural selection into an important set of questions about natural classification that had been discussed at some length by scientists of the period. Like Whewell, Darwin thinks we need more than mere resemblance to establish the natural system. On Whewell's proposal the 'something more' is the natural affinities of organisms as whole organic systems. On Darwin's proposal the 'something more' is propinquity of descent. Despite the differences, there are very important similarities: they both point out that resemblance accounts eventually must start distinguishing important functions from unimportant ones, and they both hold that the only way to do this in a non-arbitrary way is to appeal to constancies based on conditions of existence. Because they have different proposals, the constancies they appeal to are slightly different -- Whewell's constancy is internal, whereas Darwin's is comparative. But their criticisms of resemblance-only accounts are kin.
It's possible, in fact, to see Darwin's proposal as simply a larger proposal that embraces Whewell's own -- i.e., to see Darwin as taking Whewell's position and subsuming it. Darwin explicitly proposes descent with modification as the explanation for natural affinity:
All the foregoing rules and aids and difficulties in classification are explained, if I do not greatly deceive myself, on the view that the natural system is founded on descent with modification; that the characters which naturalists consider as showing true affinity between any two or more species, are those which have been inherited from a common parent, and, in so far, all true classification is genealogical; that community of descent is the hidden bond which naturalists have been unconsciously seeking, and not some unknown plan of creation, or the enunciation of general propositions, and the mere putting together and separating objects more or less alike.
That is, insofar as natural affinity is a matter of arrangement, it is a matter of propinquity of descent. (Insofar as it is a matter of closeness of resemblance, it is dependent on the contingencies of modification.) Darwin points out that, to an extent, naturalists have to bring propinquity of descent into the natural system anyway: species by their very definition have propinquity of descent. So Darwin is not proposing something completely alien to the naturalist's attempt to classify: he is proposing that naturalists expand something they have always done. Indeed, he goes farther than that, suggesting that unconsciously naturalists often appealed to descent for much of the rest of their classification. Like the vague conception of natural affinity we find Whewell attributing to natural historians, this unconscious appeal to descent serves both to connect the proposed account to actual natural-historical practice and, perhaps more interestingly, is said to account for the actual results of natural history. In other words, just as Whewell argues that the classifications of natural historians are already based on natural affinity, however vague and confused the conception of it sometimes has been, so Darwin argues that they are already based on descent, however 'unconscious' and confused its role in the classification has been:
I believe this element of descent is the hidden bond of connexion which naturalists have sought under the term of the Natural System. On this idea of the natural system being, in so far as it has been perfected, genealogical in its arrangement, with the grades of difference between the descendants from a common parent, expressed by the terms genera, families, orders, &c., we can understand the rules which we are compelled to follow in our classification. We can understand why we value certain resemblances far more than others; why we are permitted to use rudimentary and useless organs, or others of trifling physiological importance; why, in comparing one group with a distinct group, we summarily reject analogical or adaptive characters, and yet use these same characters within the limits of the same group. We can clearly see how it is that all living and extinct forms can be grouped together in one great system; and how the several members of each class are connected together by the most complex and radiating lines of affinities.
In comparing Darwin's view with Whewell, I don't want to imply that Darwin is necessarily thinking of Whewell (among others) in this argument, although that is very possible. But Whewell, as is fitting the father of the philosophy of science, is very careful to make base his philosophy of the classificatory sciences on things that had been learned and proposed by natural historians themselves; although he clarifies, refines, and develops ideas, he is primarily cleaning up in order to show clearly the state of the field in his time. Whewell is able to bring classification in natural history up to natural affinity; Darwin goes farther, and proposes to explain even natural affinity by descent with modification. In so doing he sets the notion of natural classification in biology on a new basis. (Of course, it is important to remember that biological natural history is not the only classificatory science. Whewell had a special interest in mineralogy, and, in fact, much of his actual scientific work was in improving mineralogical classification. As a general account of natural systems, Whewell's Idea of Natural Affinity is still necessary; descent with modification only explains natural affinity for natural systems that are genealogical in origin.)
For a discussion of how natural classification was understood by nineteenth-century natural historians after Darwin, see Robert O'Hara's paper on the subject.