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All the species of organisms on Earth can be connected in a grand Tree of Life. To construct such a tree, you need to measure a number of traits on the entire sample of species. Then, using some algorithm, you cluster species according to who is most similar to whom for these traits. The tree that your algorithm gives you tells you not only who is most closely related to whom, but also the history of lineage splitting events that gave rise to the species we observe today.
In the tree in the figure, for instance, we can start at the top, where all of the branches meet at a node. This node represents a species that is the most recent common ancestor of the species ‘a’–‘i.’ For scale, let’s say that it was 100 million years ago when that ancestral species split into two separate species, creating two lineages whose descendants live to this day. If we follow the lineage that descended through time on the left, we see that it also split into two further lineages about 50 million years ago. One of these lineages, at left, leads to the present-day ‘a’ species. The other lineage descends through time but splits again ~40 million years ago and each of those subsequent lineages has descendants in the present day: one branch leads to ‘b’ and the other leads to ‘e.’ That is the historical interpretation of this tree. We can also interpret this tree to say that ‘b’ and ‘e’ are more closely related to each other evolutionarily than either is to species ‘a.’
An analogy that we often make for the historical pattern that these trees depict is to our own family trees. We might say:
“Consider your grandfather, Grandpa Willy, who was born 100 years ago. By age fifty, he had two sons – Nat and Pat, who each had two daughters recently. Nat begat Natalie and Natalina, and Pat begat Patricia and Patromina.”
Starting at Grandpa Willy, we can draw something like the lineage splitting events I mentioned above for the species. In the analogy, we have a split of the lineage into two descendant lineages every generation, and we had something like that in the species tree example: one lineage splits into two, which split into more, etc. This family tree analogy is a useful one in that it gives us a sense that a sample of current species has some common ancestor back in time and that as time goes forward, lineages split to form new species. But otherwise this is a lousy analogy, for a number of reasons.
The first of these reasons would be brought to our attention by Grandma Vickie. Or, for that matter, by Grandpappy Joe or Nana Gertrude. Natalie, the living descendant of Grandpa Willy actually has four ancestors going back 100 years. In contrast, species ‘b’ goes back along just one single lineage to any of its ancestors. As we trace the ancestry of people, we accumulate ancestors with each generation farther back in time; not so for species.
The second reason highlights a philosophical problem for people who build these trees or think about these issues. Natalie and Natalina have a most recent common ancestor – it’s Nat. Species ‘b’ and species ‘e’ likewise have a common ancestor, which existed ~40 million years ago. But what is the identity of that ancestral species? This was the question that student asked and it turns out to be a much more difficult question for species trees than it is for human family trees.
We can start by asking: Was that common ancestor either ‘b’ or ‘e’ perhaps? The answer to this question is “probably not.” A lot of evolution can happen in 40 million years, and if we took a time machine back that long ago, we probably wouldn’t recognize the ancestral species as either ‘b’ or ‘e.’ Generally speaking, we don’t say that the ancestor of two species was either one of them, though there is a tendency to mistakenly portray it this way when it comes to our own species. I’ve heard many people say that we evolved from chimps about six million years ago. What they’re suggesting is that the common ancestor of chimps and humans was a chimp. But none of the species that we recover from the fossil record from that time is identified as the present-day chimp. Humans didn’t evolve from chimps and chimps didn’t evolve from humans. Both evolved from a common ancestor. But we still haven’t put an identity on that ancestor.
Let’s next ask: Was that ancestral species a third species that we haven’t yet named? The answer to this question is, “probably yes.” For the ancestor of ‘b’ and ‘e’ and for the ancestor of humans and chimps, it is likely that a time machine would back up this answer. Given that much time since common ancestry, both lineages will likely diverge away from and cease to perfectly resemble the ancestral species from which they arose. But surprisingly, and perhaps troubling, many biologists would stick with this answer over even shorter time scales. According to them, for instance, species ‘j’ undergoes speciation to give rise to two new species, ‘k’ and ‘l’ – neither of which is ‘j’ because ‘j’ ceases to exist once speciation happens. Thus, one species (‘j’) gives rise to two brand new species (‘k’ and ‘l’) at the point of speciation, with the original one (‘j’) ceasing to exist. There are few paleontologists who would put forward an argument like this, though. This is because paleontologists can follow a single species like ‘j’ through time in a column of rock. Let's say we look at a series of fossils of 'j' going through time and that halfway through its existence, it gives rise to a new species, 'k’ (see the depiction of fossil lineages below, and note that time goes in the other direction compared with the first figure). The paleontologists don’t say that ‘j’ went extinct at that point just because the new species came into existence. They can see from the fossil record that ‘j’ persisted after the new species arose. Earlier there is just ‘j,’ and then later we have both ‘j’ and ‘k.' Many “neontologists” (a term paleontologists often use for distinction) instead add a third species to the mix here, seemingly out of thin air, because ‘j’ has to go extinct under their paradigm.

This is just one way to highlight the vagueness of the species concept. There is no remedy for this vagueness, much as there is no point at which a few grains of sand becomes a heap.
:)
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