**Core Concept**
The quinsy, also known as a peritonsillar abscess, is a complication of tonsillitis where a collection of pus forms beside the tonsils. Surgical intervention in this area requires careful consideration of the surrounding anatomical structures, particularly the blood vessels.

**Why the Correct Answer is Right**
During surgical removal of the pathologic tissue or incision and drainage of the peritonsillar abscess, the **Artery of the tonsillar pillar** (also known as the tonsillar branch of the facial artery) is at greatest risk of injury. This artery supplies blood to the palatine tonsil and the surrounding tissues. In the context of quinsy, the artery of the tonsillar pillar is particularly vulnerable due to its proximity to the abscess.

**Why Each Wrong Option is Incorrect**
* **Option A:** The lingual artery primarily supplies the tongue and is not directly related to the quinsy or the surgical site.
* **Option B:** The maxillary artery is a terminal branch of the external carotid artery, which supplies blood to the maxillary sinus and surrounding structures, but it is not as directly involved in the surgical site as the artery of the tonsillar pillar.
* **Option C:** The external carotid artery is a major artery that supplies blood to the face and neck, but it is not the specific artery at greatest risk during surgical removal of the pathologic tissue or incision and drainage of the peritonsillar abscess.

**Clinical Pearl / High-Yield Fact**
Remember that the artery of the tonsillar pillar is a critical structure to identify and preserve during surgical procedures involving the tonsils, as injury to this artery can lead to significant bleeding and complications.

**Correct Answer:** C. External carotid artery