The constituency problem, as Jangari correctly noted, concerns the prepositional phrase "with the solemn precision of scientists articulating chemical equations." The original diagramming indicates that the "we" of the sentence are learning with precision, but it seems much more natural to assume that it is the diagramming that occurs with precision. In other words, the PP modifies diagram, not learned.
The second problem involves the treatment of the infinitive "to diagram..." It is treated as a modifier of the verb learned, but in the traditional analysis that Reed-Kellogg diagrams follow, it should be its direct object. (This analysis draws a parallel between instances like "We learned our lesson", where our lesson is clearly a direct object, and those like "We learned to diagram.")
Interestingly, though, the Cambridge Grammar of the English Language makes the case that infinitives and other non-noun-phrase complements of verbs should not be treated as direct objects. They are complements, yes, but not direct objects. If you accept that analysis, then I suppose you could argue that the constituency is correctly shown in the original diagram, as it is parallel to the way an infinitive is drawn when it is the complement of another part of speech (as in "eager to please"). The only problem with that argument is that other diagrams in the book stick to the traditional analysis. Further, the CGEL's analysis of constituency runs contrary to many other aspects of Reed-Kellogg diagramming, and if you're going to start to import ideas from modern linguistics, R-K diagrams would require massive revisions in many other aspects. In short, I doubt very much that this particular diagramming decision was intentional.
I find these problems interesting not so much for their "gotcha" value as for the light they shed on the utility of Reed-Kellogg diagrams. Here we have a diagram of a sentence that is not particularly complex, one that presents no real interpretive difficulties within the Reed-Kellogg system. Presumably the book's author, her sentence diagramming consultant, Gene Moutoux, and perhaps an editor or two looked the diagram over, and yet no one caught these fairly glaring problems.
I can only conclude that no one who scrutinized the diagram before it was published really thought deeply about the assertions it was making as to the sentence structure. Once the diagram was drawn, once the diagrammer's immediate job of parsing was complete, it ceased to become a serious statement of grammatical structure and became a kind of geeky, abstract artwork, intended to be admired for its peculiar arrangement of lines.
Here, for those of you who are interested, is a corrected diagram that follows the usual R-K rules. For those of you who find such diagrams aesthetically displeasing, I challenge you to create your own, more graceful diagramming system.