 It may be doubted whether any man ever possessed a more acute and perfect intellect than that of Blaise Pascal. He was born in 1623 at Clermont in Orvern, and from his earliest years displayed signs of a remarkable character. His father attempted at first to prevent his studying geometry, but such was Pascal's genius and love of this science that by the age of twelve he had found out many of the propositions of Euclid's first book without the aid of any person or treatise. It is difficult to say whether he is most to be admired for his mathematical discoveries, his invention of the first calculating machine, his wonderful provincial letters written against the Jesuits, or for his profound pensets or thoughts, a collection of his reflections on scientific and religious topics. Among these thoughts is to be found a remarkable fragment upon logical method, the substance of which is also given in the Port Royal Logic. It forms the second article of the pensets. As I know no composition in which perfection of truth and clearness of expression are more nearly attained, I propose to give in this lesson a free translation of the more important parts of this fragment, appending to it rules of method from the Port Royal Logic. The words of Pascal are nearly as follows. The true method which would furnish demonstrations of the highest excellence, if it were possible to employ the method fully, consists in observing two principle rules. The first rule is not to employ any term of which we have not clearly explained the meaning. The second rule is never to put forward any proposition which we cannot demonstrate by truths already known, that is to say in a word, to define all the terms and to prove all the propositions. But in order that I may observe the rules of the method which I am explaining, it is necessary that I declare what is to be understood by definition. We recognise in geometry only those definitions which logicians call nominal definitions, that is to say only those definitions which impose a name upon things clearly designated in terms perfectly known, and I speak only of those definitions. The value and use is to clear and abbreviate discourse, by expressing in the single name which we impose what could not be otherwise expressed but in several words, provided nevertheless that the name imposed remained divested of any other meaning which it might possess, so as to bear that alone for which we intend it to stand. For example if we need to distinguish among numbers those which are divisible into two parts from those which are not so divisible, in order to avoid the frequent repetition of this distinction we give a name to it in this manner. We call every number divisible into two equal parts an even number. This is a geometrical definition because after having clearly designated a thing, namely any number divisible into two equal parts, we give it a name divested of every other meaning which it might have, in order to bestow upon it the meaning designated. Hence it appears that definitions are very free and that they can never be subject to contradiction for there is nothing more allowable than to give any name we wish to a thing which we have clearly pointed out. It is only necessary to take care that we do not abuse this liberty of imposing names by giving the same name to two different things. Even that would be allowable provided we did not confuse the results and extend them from one to the other. But if we fall into this vice we have a very sure and infallible remedy. It is to substitute mentally the definition in place of the thing defined and to hold the definition always so present in the mind that every time we speak for instance of an even number we may understand precisely that it is a number divisible into two equal parts and so that these two things should be so combined and inseparable in thought that as often as one is expressed in discourse the mind may direct itself immediately to the other. Geometers and all who proceed methodically only impose names upon things in order to abbreviate discourse and not to lessen or change the ideas of the things concerning which they discourse. They pretend that the mind always supplies the entire definition of the brief terms which they employ simply to avoid the confusion produced by a multitude of words. Nothing prevents more promptly and effectively the insidious fallacies of the soffists than this method which we should always employ and which alone suffices to banish all sorts of difficulties and equivocations. These things being well understood I return to my explanation of the true method which consists as I said in defining everything and proving everything. Certainly this method would be an excellent one were it not absolutely impossible. It is evident that the first terms we wish to define would require previous terms to serve for their explanation and similarly the first propositions we wish to prove would presuppose other propositions preceding them in our knowledge and thus it is clear that we should never arrive at the first terms or first propositions. Accordingly in pushing our researches further and further we arrive necessarily at primitive words which we cannot define and at principles so clear that we cannot find any principles more clear to prove them by. Thus it appears that men are naturally and inevitably incapable of treating any science whatever in a perfect method but it does not thence follow that we ought to abandon every kind of method. The most perfect method available to men consists not in defining everything and demonstrating everything nor in defining nothing and demonstrating nothing but in pursuing the middle course of not defining things which are clear and understood by all persons but of defining all others and of not proving truths known to all persons but of proving all others. From this method they equally err who undertake to define and prove everything and they who neglect to do it in things which are not self-evident. It is plain in this admirable passage that we can never by using words avoid an ultimate appeal to things because each definition of a word must require one or more other words which also will require definition and so on ad infinitum. Nor must we ever return back upon the words already defined for if we define A by B and B by C and C by D and then D by A we commit what may be called a circular indefiniendo a most serious fallacy which might lead us to suppose that we know the nature of A, B, C and D when we really know nothing about them. Pascal's views of the geometrical method were clearly summed up in the following rules inserted by him in the Port Royal logic. One, to admit no terms in the least obscure or equivocal without defining them. Two, to employ in the definitions only terms perfectly known or already explained. Three, to demand as axioms only truths perfectly evident. Four, to prove all propositions which are at all obscure by employing in their proof only the definitions which have preceded or the axioms which have been accorded or the propositions which have been already demonstrated or the construction of the thing itself which is in dispute when there may be any operation to perform. Five, never to abuse the equivocation of terms by failing to substitute for them mentally the definitions which restrict and explain them. The reader will easily see that these rules are much more easy to lay down than to observe since even geometers are not agreed as to the simplest axioms to assume or the best definitions to make. There are many different opinions as to the true definition of parallel lines and the simplest assumptions concerning their nature and how much greater must be the difficulty observing Pascal's rules with confidence in less certain branches of science. Next after geometry mechanics is perhaps the most perfect science yet the best authorities have been far from agreeing as to the exact definitions of such notions as force, mass, moment, power, inertia and the most different opinions are still held as to the simplest axioms by which the law of the composition of forces may be proved. Nevertheless if we steadily bear in mind in studying each science the necessity of defining every term as far as possible and proving each proposition which can be proved by a simpler one we should do much to clear away error and confusion.