mediate Perception of the Mind, than an Attention to the Ideas compared. For from the very Genefis of a Circle it is plain, that the Circumference is every where diftant from the Center, by the exact Length of the defcribing Line; and that the feveral Radii are in Truth nothing more, than one and the fame Line variously pofited within the Figure. This short Defcription will I hope ferve, to give fome little Infight into the Manner of deducing Mathematical Principles, as well as into. the Nature of that Evidence which accompanies them. Propofitions divided into Speculatine and practical. VIII. AND now I proceed to obferve, that in all Propofitions, we either affirm or deny fome Property of the Idea that conftitutes the Subject of our Judgment, or we maintain that fomething may be done or effected. The firft Sort are called fpeculative Propofitions, as in the Example mentioned above, the Radii of the fame Circle are all equal one to another. The others are called practical, for a Reason too obvious to be mentioned; thus, that a right Line may be drawn from one Point to another, is a practical Propofition; inasmuch as it expreffes that fomething may be done. Hence Mathe matical Prin ciples diflinguished into Axioms and Poftulares. IX. FROM this twofold Confideration of Propofitions, arifes the twofold Divifion of Mathematical Principles, into Axioms and Poftulates. By an Axiom they underftand any felf-evident fpeculative Truth; as that the whole is greater than its Parts: That Things equal to one and the fame Thing, are equal to one another. But a felf-evident practical Propofition is what they call a Poftulate. Such are those of Euclid; that a finite Right-Line may be continued directly forwards: That a Circle may be defcribed about any Center with any Diflance. And here we are to obferve, that as in an Axiom, the Agreement or Difagreement between the Subject and Predicate, must come under the immediate Infpection of the Mind; fo in a Poftulate, not only the Poffibility of the Thing afferted, must be evident at first View, but alfo the Manner in which it may be effected. For where this Manner is not of itself apparent, the Propofition comes under the Notion of the demonitrable kind, and is treated as fuch by Geometrical Writers. Thus, to draw a Right-Line from one Point to another; is affumed by Euclid as a Poftulate, because the Manner of doing it is fo obvious, as to require no previous Teaching. But then it is not equally evident, how we are to conftruct an equilateral Triangle. For this Reafon he advances it as a demonftrable Propofition, lays down Rules for the exact Perfor mance, mance, and at the fame time proves, that if these Rules are followed, the Figure will be justly defcribed. I And demon Arable Propo fitions into teorems and. Problems. X. This naturally leads me to take notice, that as felf-evident Truths are diftinguished into different Kinds, according as they are speculative or practical; fo is it also with demonftrable Propofitions. A demonftrable fpeculative Propofition, is by Mathematicians called a Theorem. Such is the famous 47th Propofition of the first Book of the Elements, known by the Name of the Pythagoric Theorem, from its fuppofed Inventor Pythagoras, viz. That in every right-angled Triangle, the Square defcribed upon the Side fubtending the Right-Angle, is equal to both the Squares defcribed upon the Sides containing the Right-Angle. On the other hand, a demonftrable practical Propofition, is called a Problem; as where Euclid teaches us, to defcribe a Square upon a given Right-Line way Corollaries are obvious De. Theorems or XI. SINCE I am upon this Subject, it may not be amifs to add, that besides the four Kinds of Propofitions already mentioned, Mathematicians have ductions from alfo a fifth, known by the Name of Corollaries. Thefe are ufually fubjoined to Theorems, or Problems, and differ from them only in this; that they flow from what is there demonftrated in fo obvious a Manner, as to difcover their Dependence upon the Propofition whence they are deduced, almoft as foon as propofed. Thus Euclid having demonftrated, that in every right-lined Triangle, all the three Angles taken together are equal to two Right-Angles; adds by of Corollary, that all the three Angles of any one Triangle taken together, are equal to all the three Angles of any other Triangle taken together: which is evident at first Sight; becaufe in all Cafes they are equal to two right ones, and Things equal to one and the fame thing, are equal to one another, XII. THE laft Thing I fhall take notice of in the Practice of the Mathematicians, is what they call their Scholia. They are indifferently annexed to Definitions, Propofitions, or Corollaries; and anfwer the fame Purposes as Annotations upon a Claffic Author. For in them Occafion is taken, to explain whatever may appear intricate and obfcure in a Train of Reasoning; to anfwer Objections; to teach the Applicacation and Ufes of Propofitions; to lay open the Original and Hiftory of the feveral Discoveries made in the Science; and in a word, to acquaint us with all fuch Particulars as deferve to be known, whether confidered as Points of Curiofity or Profit, XIII. THU H 3 Sebolia ferve the Purpofes of Annota tions or d Com ment, This Methed of the Mathematicians uni. verfal, and a fure Guide to Certainty. XIII. THUS we have taken a fhort View of the fo much celebrated Method of the Mathematicians; which to any one who confiders it with a proper Attention, muft needs appear univerfal, and equally applicable in other Sciences. They begin with Definitions. From these they deduce their Axioms and Poftulates, which ferve as Principles of Reafoning; and having thus laid a firm Foundation, advance to Theorems and Problems, eftablishing all by the ftrictest Rules of Demonftration. The Corollaries flow naturally and of themfelves. And if any Particulars are ftill wanting, to illustrate a Subject, or compleat the Reader's Information; these, that the Series of Reafoning may not be interrupted or broken, are generally thrown into Scholia. In a Syftem of Knowledge fo uniform and well connected, no wonder if we meet with Certainty; and if thofe Clouds and Darkneffes, that deface other Parts of human Science, and bring Difcredit even upon Reafon itself, are here fcattered and difappear. Self-evident Truths known by the appa rent unavoidable Connec tion, between the Subject and Predicate. XIV. BUT I fhall for the prefent wave these Reflections, which every Reader of Understanding is able to make of himself, and return to the Confideration of felf-evident Propofitions. It will doubtlefs be expected, after what has been here faid of them, that I should establish some Criteria or Marks, by which they may be diftinguifhed. But I frankly own my Inability in this refpect, as not being able to conceive any thing in them, more obvious and ftriking, than that Self-evidence which constitutes their very Nature. All I have therefore to obferve on this Head is, that we ought to make it our first Care, to obtain clear and determinate Ideas. When afterwards we come to compare thefe together, if we perceive between any of them a neceflary and unavoidable Connection, infomuch that it is impoffible to conceive them exifting afunder, without destroying the very Ideas compared; we may then conclude, that the Propofition exprefing this Relation, is a Principle, and of the kind we call feli-evident. In the Example mentioned above, The Radii of the fame Circle are all equal between themselves, this intuitive Evidence thines forth in the cleareft manner; it being impoffible for any one who attends to his own Ideas, not to perceive the Equality here afferted. For as the Circumference is every where diftant from the Center, by the exac Length of the defcribing Line; the Radii drawn from the Center into the Circumference, being feverally equal to this one Line, muft needs alfo be equal among themfelves. If we fuppofe the the Radii unequal, we at the fame time fuppofe the Circumference more diftant from the Center in fome Places than in others; which Suppofition, as it would exhibit a Figure quite different from a Circle, we fee there is no feparating the Predicate from the Subject in this Propofition, without destroying the Idea, in relation to which the Comparison was made. The fame thing will be found to hold, in all our other intuitive Perceptions, infomuch that we may eftablish this as an univerfal Criterion, whereby to judge of and distinguish them. I would not however be understood to mean, as if this ready View of the unavoidable Connection between fome Ideas, was any thing really different from Self-evidence. It is indeed nothing more than the Notion of Self-evidence a little unfolded, and as it were laid open to the Infpection of the Mind. Intuitive Judgments need no other diftinguishing Marks, than that Brightnefs which furrounds them; in like manner as Light difcovers itfelf by its own Prefence, and the Splendor it univerfally diffules. But I have faid enough of felf-evident Propofitions, and fhall therefore now proceed to thofe of the demonftrable kind; which being gained in Confequence of Reafoning, naturally leads us to the third Part of Logick, where this Operation of the Understanding is explained. Of Reafoning in general, and the Parts of which it confifts. Remate Rela I. tions discover intermediate Ideas. WE have feen how the Mind proceeds in furnishing itself with Ideas, and framing ed by means of intuitive Perceptions. Let us next enquire into the Manner of difcovering thofe more remote Relations, which lying at a Distance from the Understanding, are not to be traced, but by means of a higher Exercife of its Powers. It often happens in comparing Ideas together, that their Agreement or Dilagreement cannot be difcerned at firft Vicw, efpecially if they are of fuch a Nature, as not to admit of an exact Application one to another. When for inftance we compare two Figures of a different Make, in order to judge of their Equality or Inequality, it is plain, that by barely confidering the Figures themfelves, we cannot arrive at an exact Determination; becaufe by reafon of their difagreeing Forms, it is impoffible fo to put them together, as that their feveral Parts fhall mutually coincide. Here then it becomes |