diff --git a/elastica/external_forces.py b/elastica/external_forces.py
index 6d377acb..d7f5185e 100644
--- a/elastica/external_forces.py
+++ b/elastica/external_forces.py
@@ -508,8 +508,12 @@ def __init__(
         start_force_mag: float,
         end_force_mag: float,
         ramp_up_time: float = 0.0,
-        tangent_direction: NDArray[np.float64] = np.array([0, 0, 1]),
-        normal_direction: NDArray[np.float64] = np.array([0, 1, 0]),
+        tangent_direction: NDArray[np.floating] = np.array(
+            [0.0, 0.0, 1.0]
+        ),  # FIXME: avoid mutable default
+        normal_direction: NDArray[np.floating] = np.array(
+            [0.0, 1.0, 0.0]
+        ),  # FIXME: avoid mutable default
     ) -> None:
         """
 
diff --git a/elastica/modules/protocol.py b/elastica/modules/protocol.py
index b9a52ec7..4ca93a17 100644
--- a/elastica/modules/protocol.py
+++ b/elastica/modules/protocol.py
@@ -1,5 +1,5 @@
 from typing import Protocol, Generator, TypeVar, Any, Type, overload
-from typing_extensions import Self  # 3.11: from typing import Self
+from typing_extensions import Self  # python 3.11: from typing import Self
 
 from abc import abstractmethod
 
diff --git a/elastica/rod/cosserat_rod.py b/elastica/rod/cosserat_rod.py
index 8ac1d081..1027d226 100644
--- a/elastica/rod/cosserat_rod.py
+++ b/elastica/rod/cosserat_rod.py
@@ -188,7 +188,7 @@ def __init__(
         internal_couple: NDArray[np.float64],
         ring_rod_flag: bool,
     ) -> None:
-        self.n_nodes = n_elements + 1 if ring_rod_flag else n_elements
+        self.n_nodes = n_elements + 1 if not ring_rod_flag else n_elements
         self.n_elems = n_elements
         self.position_collection = position
         self.velocity_collection = velocity
diff --git a/elastica/typing.py b/elastica/typing.py
index 07ac71f5..1ba1a4ee 100644
--- a/elastica/typing.py
+++ b/elastica/typing.py
@@ -40,7 +40,6 @@
 SystemIdxType: TypeAlias = int
 BlockSystemType: TypeAlias = "BlockSystemProtocol"
 
-#    NoForces | NoContact | FreeJoint | FreeBC | DamperBase | CallBackBaseClass
 
 # Mostly used in explicit stepper: for symplectic, use kinetic and dynamic state
 StateType: TypeAlias = "State"