This paper introduces the inerter, a new mechanical circuit element that serves as the true dual of the electrical capacitor. The inerter allows for the synthesis of mechanical networks using classical electrical circuit synthesis methods, addressing the limitations of traditional mechanical analogies. The inerter is designed to have negligible mass and finite linear travel, making it suitable for various applications such as vibration absorption, suspension strut design, and simulating mass elements. The paper discusses the force-current analogy between electrical and mechanical networks, and provides a detailed definition and properties of the inerter. It also presents classical network synthesis results for mechanical networks, including theorems on the synthesis of positive-real functions. The inerter is applied to solve vibration absorption problems, demonstrating its effectiveness compared to conventional approaches. Additionally, the inerter is used to design suspension struts, showing improved performance in terms of static stiffness and damping ratio. The paper concludes with a numerical example and a discussion on the realization of the inerter using Brune and Darlington synthesis methods.This paper introduces the inerter, a new mechanical circuit element that serves as the true dual of the electrical capacitor. The inerter allows for the synthesis of mechanical networks using classical electrical circuit synthesis methods, addressing the limitations of traditional mechanical analogies. The inerter is designed to have negligible mass and finite linear travel, making it suitable for various applications such as vibration absorption, suspension strut design, and simulating mass elements. The paper discusses the force-current analogy between electrical and mechanical networks, and provides a detailed definition and properties of the inerter. It also presents classical network synthesis results for mechanical networks, including theorems on the synthesis of positive-real functions. The inerter is applied to solve vibration absorption problems, demonstrating its effectiveness compared to conventional approaches. Additionally, the inerter is used to design suspension struts, showing improved performance in terms of static stiffness and damping ratio. The paper concludes with a numerical example and a discussion on the realization of the inerter using Brune and Darlington synthesis methods.