To abridge that species of mental labour which is required in conducting arithmetical computations, has been the professed object of a variety of mechanical contrivances. But the greater number of arithmetical machines, as they have been called, are more ingenious than really useful, and have been recorded more as objects of curiosity, than as admitting of convenient or ready application in the actual practice of arithmetic. The machine invented by Pascal, and others constructed on the same principle, were, strictly speaking, limited to the simpler operations of addition and subtraction, and were incapable of being applied to the finding of products or quotients in any other way than by effecting a number of successive additions or subtractions. Still less did they aim at the immediate performance of the higher operations of involution, which, even by the most compendious methods of arithmetic, is a laborious process ; or of the extraction of roots, to which the common rules furnish but a circuitous and slow approximation. The only instruments which promise to afford real assistrance to the practical calculator, are those founded on the theory of logarithms: a theory, which has been the fertile source, as well as the commodious instrument of discovery in every department of mathematical inquiry. The scale of Gunter, and the common sliding-rule, are derived from the properties of logarithms; and the purposes to which they are immediately applicable are the multiplication and division of numbers. The instrument, of which I purpose giving an account to the Society in this communication, is founded on a particular mode of employing logarithms, and is calculated to apply immediately to the involution and evolution of numbers. To those who are already conversant with mathematical pursuits, a few words would suffice to explain the principle on which it operates: but to such as are not familiar with the practical employment of logarithms, or of the common sliding-rule, the following statement of the chain of reasoning on which they depend, may conduce to render the subsequent details more intelligible.