as little doubt that he will rife to-morrow, as that he is now fet. There are many other propofitions, the truch of which is probable only, not abfolutely certain; as, for example, that things will continue in their ordinary ftate. That natural operations are performed in the simplest manner, is an axiom of natural philofophy: it may be probable, but is far from being certain *. In every one of the inftances given, conviction arifes from a fingle act of perception: for which reafon, knowledge acquired by means of that perception, not only knowledge in its proper sense, but also opinion and belief, are termed intuitive knowledge. But there are many things, the knowledge of which is not obtain- . ed with fo much facility. Propofitions for the most part require a procefs or operation in the mind, termed reafoning; leading, by certain intermediate fteps, to the propofition that is to be demonftrated or made evident; which, in oppofition to intuitive knowledge, is termed difcurfive knowledge. This process or operation must be explained, in order to understand the nature of reafoning. And as reasoning is moftly employ'd in difcovering relations, I fhall draw my examples from them. Every propofition concerning relations, is an affirmation of a certain relation between two fubjects. If the relation affirmed appear not intuitively, we must fearch for a third fubject, that appears intuitively to be connected with each of the others, by the relation affirmed: and if fuch a fubject be found, the propofition is demonftrated; for it is I have given this propofition a place, because it is affumed as an axiom by all writers on natural philofophy. And yet there appears fome room for doubting, whether the conviction we have of it do not proceed from a bias in our nature, rather than from an original fenfe. Our tafte for fimplicity, which undoubtedly is natural, renders fimple operations more agreeable than what are complex, and confequently makes them appear more natural. It deferves a moft ferious difcuffion, whether the operations of nature be always carried on with the greatest fimplicity, or whether we be not misled by our tafte for fimplicity, to be of that opinion. intuitively certain, that two fubjects, connected with a third by any particular relation, must be connected together by the fame relation: The longest chain of reafoning may be linked together in this manner. Running over fuch a chain, every one of the fubjects must appear intuitively to be connected with that imme diately preceding, and with that immediately fubfequent, by the relation affirmed in the propofition; and from the whole united, the propofition, as above mentioned, muft appear intuitively cer tain. The last step of the process is termed a conclufion, being the last or concluding perception. No fort of reafoning affords fo clear a notion of the foregoing procefs, as that which is mathematical. Equality is the only inathematical relation; and comparifon therefore is the only means by which mathematical propofitions are afcertained. To that fcience belong a fet of intuitive propofitions, termed axioms, which are all founded ons equality. For example: Divide two equal lines, each of them, into a thousand equal parts, a fingle part of the one line must be equal to a fingle part of the other Second: Take ten of thefe parts from the one line, and as many from the other, and the remaining parts must be equal: which is more fhortly expreffed thus: From two equal lines take equal parts, and the remainders will be equale; or add equal parts, and the fums will be equal. Third: If two things be, in the fame refpect, equal to a third, the one is equal to the other in the fame refpect. I proceed to fhow the ufe of thefe axioms. Two things may be equal without being intuitively for; which is the cafe of the equality between the three angles of a triangle and two right angles. To demonftrate that truth, it is neceflary to fearch for fome other angles, which appear by intuition to be equal to both. If this property cannot be discovered in any one fet of angles, we must go more leisurely to work, by trying to find angles that are equal to the three angles of a triangle.Thefe being difcovered, we next try to find other angles equal to the i angles angles now difcovered; and fo on in the comparifon, till at last we discover a fet of angles, equal not only to thofe thus introdu ced, but also to two right angles. We thus connect the two parts of the original proposition, by a number of intermediate equali ties; and by that means perceive, that these two parts are equal among themselves; it being an intuitive propofition, as mentioned above, That two things are equal, each of which, in the fame refpect, is equal to a thirdə bulan 973. F I proceed to a different example, which concerns the relation between cause and effect. The proposition to be demonstrated is, "That there exists a good and intelligent Being, who is the cause "of all the wife and benevolent effects that are produced in the government of this world." That there are fuch effects, is in the prefent example the fundamental propofition, which is taken for granted, because it is verified by experience. In order to difcover the cause of these effects, begin with an intuitive propofition mentioned above, "That every effect adapted to a good end or purpose, proceeds from a designing and benevolent cause." The next step is, to examine whether man can be the caufe: he is provided indeed with fome fhare of wisdom and benevolence; but the effects mentioned are far above his power, and no lefs above his wifdom, Neither can this earth be the caufe, nor the fun, the moon, the ftars; for, far from being wife and benevolent, they are not even fenfible. If these be excluded, we are unavoidably led to an invifible being, endowed with boundlefs power, goodnefs, and intelligence; and that invifible being is termed God.1 t Reafoning requires two mental powers, namely, the powers of invention, and of perceiving relations. By the former are discovered intermediate propofitions, equally related to the fundamental propofition, and to the conclufion and by the latter, we perceive, that the different links which compofe the chain of reasoning, are all connected together by the fame relation.p We can reafon about matters of opinion and belief, as well as about matters of knowledge, properly fo termed. Hence reafoning is diftinguifhed into two kinds; demonstrative, and probable. Demonftrative reafoning is alfo of two kinds in the firft, the conclufion is drawn from the nature and inherent properties of the fubject: in the other, the conclufion is drawn from fome principle, of which we are certain by intuition. With refpect to the first, we have no fuch knowledge of the nature or inherent pro perties of any being, material or immaterial, as to draw conclufions from it with certainty. I except not even figure confidered as a quality of matter, tho' it is the object of mathematical reafoning. As we have no standard for determining with precifion the figure of any portion of matter, we cannot with precision reafon upon it what appears to us a ftraight line may be a curve, and what appears a rectilinear angle may be curvilinear. How then comes mathematical reafoning to be demonstrative? This question may appear at first fight puzzling; and I know not that it has any where been diftinctly explained. Perhaps what follows may be fatisfactory." The fubjects of arithmetical reafoning are numbers. The fubjects of mathematical reafoning are figures. But what figures are fubjects of mathematical reasoning? Not fuch as I fee; but fuch as I form an idea of, abftracting from every imperfection. I explain myfelf. There is a power in man to forın images of things that never exifted; a golden mountain, for example, or a river running upward. This power operates upon figures. There is perhaps no figure exifting the fides of which are ftraight lines. But it is eafy to form an idea of a line, that has no waving or crookedncfs in it; and it is easy to form an idea of a figure bounded by fuch lines. Such ideal figures are the subjects of mathematical reafoning, and thefe being perfectly clear and distinct, are proper fubjects for demonftrative reasoning of the first kind. Ma thematical thematical reasoning however is not merely a mental entertainment: it is of real ufe in life, by directing the powers and properties of matter. There poffibly may not be found any where a perfect globe, to answer the idea we form of that figure; but a globe may be made fo near perfection, as that the properties demonftrated to belong to the idea of a perfect globe will be nearly applicable to that figure. In a word, tho' ideas are, properly speaking, the subject of mathematical evidence; yet the end and purpose of that evidence is, to direct us with refpect to figures as they really exift; and the nearer any real figure approaches to the idea we form of it, with the greater accuracy will the mathematical truth be applicable. The component parts of figures, viz. lines and angles, are ex tremely fimple, requiring no definition. Place before a child a crooked line, and one that has no appearance of being crooked; call the former a crooked line, the latter a fraight line and the child will use these terms familiarly, without hazard of a miftake. Draw a perpendicular upon paper; let the child advert, that the upward line leans neither to the right nor the left, and for that reason is termed a perpendicular: the child will apply that term familiarily to a tree, to the wall of a houfe, or to any other perpendicular. In the fame manner, place before the child two lines diverging from each other, and two that have no appearance of diverging: call the latter parallel lines, and the child will have no difficulty of applying the fame term to the fides of a door. or of a window. Yet fo accuftomed are we to definitions, that even these simple ideas are not fuffered to efcape. A ftraight line, for example, is defined to be the fhorteft that can be drawn between two given points. The fact is certain; but fo far from a definition, that it is an inference drawn from the idea of a ftraight line: and had I not beforehand a clear idea of a ftraight line, I could not infer that it is the fhortest between two given points. D'Alembert |