Deferring a more complete discussion of this point, I will only mention now that arguments from double identity occur very frequently, and are usually taken for granted, owing to their extreme simplicity. In regard to the equivalence of words this form of inference must be constantly employed. If the ancient Greek χαλκός is our *copper*, then it must be the French *cuivre*, the German *kupfer*, the Latin *cuprum*, because these are words, in one sense at least, equivalent to copper. Whenever we can give two definitions or expressions for the same term, the formula applies; thus Senior defined wealth as “All those things, and those things only, which are transferable, are limited in supply, and are directly or indirectly productive of pleasure or preventive of pain.” Wealth is also equivalent to “things which have value in exchange;” hence obviously, “things which have value in exchange = all those things, and those things only, which are transferable, &c.” Two expressions for the same term are often given in the same sentence, and their equivalence implied. Thus Thomson and Tait say,[57] “The naturalist may be content to know matter as that which can be perceived by the senses, or as that which can be acted upon by or can exert force.” I take this to mean-- Matter = what can be perceived by the senses; Matter = what can be acted upon by or can exert force. [57] *Treatise on Natural Philosophy*, vol. i. p. 161. For the term “matter” in either of these identities we may substitute its equivalent given in the other definition. Elsewhere they often employ sentences of the form exemplified in the following:[58] “The integral curvature, or whole change of direction of an arc of a plane curve, is the angle through which the tangent has turned as we pass from one extremity to the other.” This sentence is certainly of the form-- The integral curvature = the whole change of direction, &c. = the angle through which the tangent has turned, &c. [58] *Treatise on Natural Philosophy*, vol. i. p. 6. Disguised cases of the same kind of inference occur throughout all sciences, and a remarkable instance is found in algebraic geometry. Mathematicians readily show that every equation of the form *y* = *mx* + *c* corresponds to or represents a straight line; it is also easily proved that the same equation is equivalent to one of the general form A*x* + B*y* + C = 0, and *vice versâ*. Hence it follows that every equation of the form in question, that is to say, every equation of the first degree, corresponds to or represents a straight line.[59] [59] Todhunter’s *Plane Co-ordinate Geometry*, chap. ii. pp. 11–14. *Inference with a Simple and a Partial Identity.* A form of reasoning somewhat different from that last considered consists in inference-between a simple and a partial identity. If we have two propositions of the forms A = B, B = BC,