mathematicks afford numerous instances of conclusions legitimately drawn from definitions and assumed principles. We also reason a priori whenever we judge of effects from a knowledge of the causes, which produce them. Thus we infer, that an eclipse of the sun and an eclipse of the moon can never happen within twelve days of each other, from our knowledge of the causes, which occasion those phenomena.

149. Reasoning a posteriori is the reverse of the former process. By this we deduce causes from effects. Thus we infer, that the earth is spherical from its shadow on the moon in a lunar eclipse; and we infer the being of a God from our own existence and that of the objects around us. All reasoning concerning the properties and laws, both of mind and body, proceeds on this principle. It is only by a careful observation of facts, that the laws, which regulate them, can be discovered.

150. Another distinction of reasoning is into direct and indirect. The reasoning is direct, when the proofs are so applied, as to show immediately the agreement or repugnancy between the subject and predicate of the proposition in

question.

In indirect reasoning, the arguments, which we employ, are not intended primarily to show the relation between the terms of the proposition, whose truth we would establish; but to prove the falsehood or absurdity of the proposition, to which it is opposed. This method may be adopted, whenever it is manifest, that the proposition, which we allege, or its contrary, must be true. We may then prove the impossibility of the contrary proposition; or we may show, that a manifest absurdity must follow from admitting it; and in either case we establish the truth of our original proposition. The former course is usually called a proof per impossibile; and the latter, a reductio ad absurdum.

131. Mathematicians make frequent use of indirect reasoning. Thus, Euclid proves, by an indirect course, that, "if two circles touch "each other internally, they cannot have the "same centre." He first supposes the contrary to be true, namely, that the two circles have the same centre; and no third supposition can be made; for they must either both have the same centre or not. He then demonstrates the impossibility of the case assumed; and thence

infers the truth of the proposition, which he

first asserted.

So moralists

So moralists prove the existence of an all-wise and powerful Creator, by tracing the absurdities, which the contrary supposition involves.

152. Another form of indirect reasoning, in frequent use, is denominated reasoning a fortiori. This consists in deducing a proposition, as true, from less obvious propositions, embraced by the same general principles. Thus, if the felon, who robs on the highway, deserves the punishment of death, this retribution is due a fortiori to the wretch, who has committed parricide.

CHAPTER NINTH.

GENERAL DESCRIPTION OF SYLLOGISTICK REASONING.

153. All reasoning proceeds by comparison; and two comparisons are necessary to enable us to make a conclusion. The subject and predicate of the proposition to be proved must be separately compared with some third term, or common measure; and from these comparisons we infer their agreement or repugnancy.

This process, when expressed in words, consists of three propositions, and has been termed syllogism.*

154. Syllogism was regarded, for many centuries, as the only sure instrument of reasoning; and skid in the use of it as the highest accomplishment, which the mind can possess. It derived its celebrity from the talents and industry of Aristotle, who traced and analyzed its principles, subjected it to laws, and exhibited it in all the variety of modes and figures, into which it could be moulded. Since the time of that philosopher, the name syllogism has usually been employed to denote an argument, framed according to certain technical rules of art. But it is sometimes used in a larger sense, to imply any process of reasoning from more general to less general, in opposition to the principle of analytical induction. In this sense, it will apply to mathematical reasoning; for all demonstrations in this science proceed on this fundamental principle of the syllogism, that whatever muy be affirmed of any genus may be affirmed of all the species included under it.

* Evdĺoyioμòs, computatio, a ovλdoyíšoμaι, colligo, ratiocinor, computv.

It

155. Syllogism and induction proceed in opposite directions. Induction, as has already been observed, begins with individual objects, as they exist in nature, and ascends by successive steps to the most general truths. Syllogism begins where induction terminates. commences with some universal proposition, and follows back the footsteps of the former process, transferring at each stage the predicate of the more general to the less general rank of beings; or, in other words, predicating the genus of the species, and the species of the individual.

156. The difference of these methods may be shown by the following example. We observe that the individual people of our acquaintance are constantly dying around us; that men rarely live to the age of a hundred years, and that the former generations are wholly swept from the earth. From these facts we infer, that death is the common lot of our species. Observing also, that the same fatality attends the various species of beasts, birds, and insects, we deduce the more general conclusion, that all animals are mortal. This inductive process, reversed in syllogistick language, would run thus,