figures, not being triangles, which have either their angles equal and sides not proportional (abCd), or their corresponding sides proportional and angles not equal (abcD), or neither their corresponding angles equal nor corresponding sides proportional (abcd)." In performing this method of inference it is soon seen to proceed in a very simple mechanical manner, and the only inconvenience is the large number of alternatives or combinations to be examined. I have, therefore, devised several modes by which the labour can be decreased; the simplest of these consists in engraving the series of 16 combinations on the opposite page, which occur over and over again in problems, with larger and smaller sets, upon a common writing slate, so that the excluded ones may be readily struck out with a common slate pencil, and yet the series may be employed again for any future logical question. A second device, which I have called the "Logical abacus," is constructed by printing the letters upon slips of wood furnished with pins, contrived so that any part or class of the combinations can be picked out mechanically with very little trouble; and a logical problem is thus solved by the hand, rather than by the head. More recently however I have reduced the system to a completely mechanical form, and have thus embodied the whole of the indirect process of inference in what may be called a Logical Machine. In the front of the machine are seen certain moveable wooden rods carrying the set of 16 combinations of letters which are seen on the preceding page. At the foot are 21 keys like those of a piano; eight keys towards the left hand are marked with the letters A, a, B, b, C, c, D, d, and are intended to represent these terms when occurring in the subject of a proposition. Eight other keys towards the right hand represent the same letters or terms when occurring in the predicate. The copula of a proposition is represented by a key in the middle of the series; the full stop by one to the extreme right, while there are two other keys which serve for the disjunctive conjunction or, according as it occurs in subject or predicate. Now if the letters be taken to stand for the terms of a syllogism or any other logical argument, and the keys of the instrument be pressed exactly in the order corresponding to the words of the premises, the 16 combinations will be so selected and arranged thereby that at the end only the possible combinations will remain in view. Any question can then be asked of the machine, and an infallible answer will be obtained from the combinations remaining. The internal construction of the machine is such, therefore, as actually to perform the work of inference which, in Dr Boole's system, was performed by a very complicated mathematical calculation. It should be added, that there is one remaining key to the extreme left which has the effect of obliterating all previous operations and restoring all the combinations to their original place, so that the machine is then ready for the performance of any new problem. An account of this logical machine may be found in the Proceedings of the Royal Society for Jan. 20th, 1870, the machine having on that day been exhibited in action to the Fellows of the Society. The principles of the method of inference here described are more completely stated in The Substitution of Similars*, and the Pure Logic†, which I published in the years 1869 and 1864. I may add, that the first-named of these works contains certain views as to the real nature of the process of inference which I do *The Substitution of Similars, the true Principle of Reasoning, derived from a modification of Aristotle's Dictum. Macmillan and Co. 1869. + Pure Logic, or the Logic of Quality apart from Quantity,&c. Edward Stanford, Charing Cross. not think it desirable to introduce into an elementary work like the present, on account of their speculative character. The process of inference, on the other hand, which I have derived from Boole's system is of so self-evident a character, and is so clearly proved to be true by its reduction to a mechanical form, that I do not hesitate to bring it to the reader's notice. George Boole, Mathematical Analysis of Logic, 1847. LESSON XXIV. ON METHOD, ANALYSIS AND SYNTHESIS. IT has been held by many writers on Logic that, in addition to the three parts of logical doctrine which treat successively of Terms, Propositions and Syllogisms, there was a fourth part, which treats of method. Just as the doctrine of Judgment considers the arranging of terms and their combination into Propositions, and the doctrine of Syllogism considers the arranging of propositions that they may form arguments, so there should in like manner be a fourth part, called Method, which should govern the arrangement of syllogisms and their combination into a complete discourse. Method is accordingly defined as consisting in such a disposition of the parts of a discourse that the whole may be most easily intelligible. The celebrated Peter Ramus, who perished in the massacre of St Bartholomew, first proposed to make method in this manner a part of the science of Logic; but it may well be doubted whether any definite set of rules or principles can be given to guide us in the arrangement of arguments. Every different discourse must consist of arguments arranged in accordance with the peculiar nature of the subject; and no general rules can be given for treating things which are infinitely various in the mode of treatment required. Accordingly the supposed general rules of method are no better than truisms, that is, they tell us nothing more than we must be supposed to know beforehand. Thus, we are instructed in composing any discourse to be careful that— 1. Nothing should be wanting or redundant. 2. The separate parts should agree with each other. Nothing should be treated unless it is suitable to the subject or purpose. 3. 4. The separate parts should be connected by suitable transitions. But it is evident that the whole difficulty consists in deciding what is wanting or redundant, suitable or consistent. Rules of this kind simply tell us to do what we ought to do, without defining what that is. There exist nevertheless certain general modes of treating any subject which can be clearly distinguished, and should be well understood by the logical student. Logic cannot teach him exactly how and when to use each kind of method, but it can teach him the natures and powers of the methods, so that he will be more likely to use them rightly. We must distinguish, The method of discovery is employed in the acquisition of knowledge, and really consists in those processes of inference and induction, by which general truths are ascertained from the collection and examination of par ticular facts. This method will be the subject of most of our remaining Lessons. The second method only applies when knowledge has already been acquired and expressed in the form of general laws, rules, principles or truths, so that we have only to make ourselves acquainted with these and observe the due mode of applying them to particular cases, in order to possess a complete acquaintance with the subject. A student, for example, in learning Latin, Greek, French, German, or any well-known language, receives a complete Grammar and Syntax setting forth the whole of the principles, rules and nature of the language. He receives these instructions, and takes them to be true on the authority of the teacher, or the writer of the book; and after rendering them familiar to his mind he has nothing to do but to combine and apply the rules in reading or composing the language. He follows, in short, the method of Instruction. But this is an entirely different and opposite process to that which the scholar must pursue who has received some writings in an unknown language, and is endeavouring to make out the alphabet, words, grammar, and syntax of the language. He possesses not the laws of grammar, but words and sentences obeying those laws, and he has to detect the laws if possible by observing their effects on the written language. He pursues, in short, the method of discovery consisting in a tedious comparison of letters, words, and phrases, such as shall disclose the more frequent combinations and forms in which they occur. The process would be a strictly inductive one, such as I shall partially exemplify in the Lessons on Induction; but it is far more difficult than the method of Instruction, and depends to a great extent on the happy use of conjecture and hypothesis, which demands a certain skill and inventive ability. Exactly the same may be said of the investigation of |