The part of a lathe machine that provides the method of holding and moving the cutting tool is the carriage.

The carriage is an essential component of a lathe machine and is responsible for the movement of the cutting tool along the workpiece. It is mounted on the bed of the lathe and can be moved longitudinally by means of a power feed or manually by an operator.

Here is a detailed explanation of the carriage and its functions:

1. Definition and Placement:
The carriage is a sliding mechanism located on the bed of the lathe machine. It moves along the bed's guideways and supports various components such as the tool post, cross-slide, and apron.

2. Tool Post:
The tool post is a component mounted on the carriage that holds the cutting tool securely in place. It allows for easy adjustment and positioning of the tool in relation to the workpiece. The tool post can be adjusted vertically and horizontally to achieve the desired cutting depth and angle.

3. Cross-Slide:
The cross-slide is another vital part of the carriage. It is positioned perpendicular to the lathe's bed and can be moved transversely to the workpiece. The cross-slide holds the tool post and enables precise movement of the cutting tool across the workpiece's diameter, allowing for shaping, grooving, and facing operations.

4. Apron:
The apron is located below the carriage and houses various essential components such as the feed rod, feed gears, and lead screw. It provides the mechanism for longitudinal movement of the carriage along the bed. The feed rod and gears allow for automatic or manual power feed, controlling the rate at which the tool moves along the workpiece.

5. Saddle:
The saddle is a major part of the carriage that supports and moves the entire assembly. It is responsible for the lateral movement of the carriage along the bed's guideways. The saddle is accurately machined and precisely fitted to ensure smooth movement and precise positioning of the carriage.

In summary, the carriage plays a crucial role in a lathe machine by providing the means to hold and move the cutting tool. It consists of the tool post, cross-slide, apron, and saddle, all working together to deliver precise and controlled tool movement for various machining operations.