Some philosophers, especially those of France, have held that the Indirect Method of Proof has a certain inferiority to the direct method, which should prevent our using it except when obliged. But there are many truths which we can prove only indirectly. We can prove that a number is a prime only by the purely indirect method of showing that it is not any of the numbers which have divisors, and the remarkable process known as Eratosthenes’ Sieve is the only mode by which we can select the prime numbers.[72] It bears a strong analogy to the indirect method here to be described. We can prove that the side and diameter of a square are incommensurable, but only in the negative or indirect manner, by showing that the contrary supposition inevitably leads to contradiction.[73] Many other demonstrations in various branches of the mathematical sciences proceed upon a like method. Now, if there is only one important truth which must be, and can only be, proved indirectly, we may say that the process is a necessary and sufficient one, and the question of its comparative excellence or usefulness is not worth discussion. As a matter of fact I believe that nearly half our logical conclusions rest upon its employment. [72] See Horsley, *Philosophical Transactions*, 1772; vol. lxii. p. 327. Montucla, *Histoire des Mathematiques*, vol. i. p. 239. *Penny Cyclopædia*, article “Eratosthenes.” [73] Euclid, Book x. Prop. 117. *Simple Illustrations.* In tracing out the powers and results of this method, we will begin with the simplest possible instance. Let us take a proposition of the common form, A = AB, say, *A Metal is an Element,* and let us investigate its full meaning. Any person who has had the least logical training, is aware that we can draw from the above proposition an apparently different one, namely, *A Not-element is a Not-metal.* While some logicians, as for instance De Morgan,[74] have considered the relation of these two propositions to be purely self-evident, and neither needing nor allowing analysis, a great many more persons, as I have observed while teaching logic, are at first unable to perceive the close connection between them. I believe that a true and complete system of logic will furnish a clear analysis of this process, which has been called *Contrapositive Conversion*; the full process is as follows:-- [74] *Philosophical Magazine*, December 1852; Fourth Series, vol. iv. p. 435, “On Indirect Demonstration.” Firstly, by the Law of Duality we know that *Not-element is either Metal or Not-metal.* If it be metal, we know that it is by the premise *an element*; we should thus be supposing that the same thing is an element and a not-element, which is in opposition to the Law of Contradiction. According to the only other alternative, then, the not-element must be a not-metal. To represent this process of inference symbolically we take the premise in the form A = AB. (1)