> Problem:   A rope wraps an angle θ around a pole. You grab one end and pull with a tension $T_0$ . The other end is attached to a large object, say, a boat. If the coefficient of static friction between the rope and the pole is $\mu$, what is the largest force the rope can exert on the boat, if the rope is not to slip around the pole?

Solution given:  
Consider a small piece of the rope that subtends an angle $dθ$. Let the
tension in this piece be $T$ (which varies slightly over the small length). The pole exerts a small outward normal force, $N_{dθ}$ , on the piece. This normal
force exists to balance the “inward” components of the tensions at the ends. These
inward components have magnitude $T \sin(dθ/2)$. 1 Therefore, $N_{dθ} = 2T \sin(dθ/2)$.
The small-angle approximation, $\sin(x) ≈ x$, allows us to write this as $N_{dθ} = T dθ$.  
The friction force on the little piece of rope satisfies $F_{dθ} ≤ μN_{dθ} = μT_{dθ}$. This
friction force is what gives rise to the difference in tension between the two ends of
the piece. In other words, the tension, as a function of θ, satisfies
$$T(\theta+d\theta)\le T(\theta) + \mu Td\theta \ \ \ (*) \\ \implies dT \le \mu Td\theta \\ \implies \int \frac{dT}{T} \le \int \mu d\theta \\ \implies \ln(t) \le \mu \theta + C \\ \implies T \le T_{0}e^{\mu \theta}$$

What I don't understand here is that the author says, the other end of the rope is "attached" to  the boat. Now this does not mean that the boat is "pulling" the rope with some tension....if that is the case (the boat pulling the rope with a large tension, say $T$), then I am clear on what has to be done, we accordingly assign a direction to the frictional force and we see that the force needed to hold the rope from slipping, i.e. $T_{0} \ge Te^{-\mu \theta}$, which is in accordance with the result given in the book.

But since the rope here is only "attached" to the boat, I don't see how the equation marked $(*)$ holds true...since "we" are "pulling" with a tension $T_{0}$, shouldn't the equation be (owing to the direction of friction..) 
$$T(\theta + d\theta) + \mu Td \theta \le T(\theta) \\ \implies T \le T_{0}e^{-\mu \theta}$$

Where am I wrong?