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 procefs or operation must be explained, in order to understand the nature of reasoning. And as reafoning is moftly employed 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, intuitively connected with each of the others by the relation affirmed and if fuch a fubject be found, the propofition is demonftrated; for it is 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 muft appear intuitively to be connected with that immediately 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 certain. The last step of the procefs is termed a conclufion, being the laft or concluding perception. No other reafoning affords fo clear a notion of the foregoing process, as that which is mathematical. Equality is the only mathematical relation; and comparison therefore is the only means by which mathematical propofitions are afcertained. To that fcience belong a number of intuitive propofitions, termed axioms, which are all founded on equality. For example: Divide two equal lines, each cach 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 twó equal lines take equal parts, and the remainders will be equal; 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 refpe&t. I proceed to show the use of these axioms. Two things may be equal without being intuitively fo; which is the cafe of the equality between the three angles of a triangle and two right angles. To demonftrate that truth, it is neceffary to fearch for fome other angles that intuitively are equal to both. If this property cannot be difcovered in any one fet of angles, we must go more leifurely to work, and try to find angles that are equal to the three angles of a triangle. Thefe being discovered, we next try to find other angles equal to the angles. now discovered; and fo on in the comparifon, till at laft we difcover a fet of angles, equal not only to those introduced, but alfo to two right angles,, We thus connect the two parts of the original propofition, by a number of intermediate equalities; and by that means perceive, that these two parts are equal among themfelves; it being an intuitive propofition, as mentioned above, That two things are equal, each of which, in the fame refpect, is equal to a third. I proceed to a different example, which concerns the relation between cause and effect. The propofition to be demonftrated is, "That there "exifts a good and intelligent Being, who is the "caufe 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 ex ample ample the fundamental propofition; which is taken for granted, because it is verified by experience. In order to difcover the caufe of thefe effects, I begin with an intuitive propofition mentioned above, "That every effect adapted to a good end "or purpose, proceeds from a defigning and be"nevolent caufe." 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 cause, 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 boundless power, goodnefs, and intelligence; and that invifible being is termed God. Reafoning requires two mental powers, namely, the power of invention, and the power of perceiving relations. By the former are difcovered intermediate propofitions, equally related to the fundamental propofition and to the conclufion: by the latter we perceive, that the different links which compofe the chain of reasoning, are all connected together by the fame relation. We can reafon about matters of opinion and belief, as well as about matters of knowledge properly fo termed. Hence reafoning is diftinguished into two kinds; demonftrative, and probable. Demonstrative reasoning is also of two kinds: in the first, the conclufion is drawn from the nature and inherent properties of the subject: in the other, the conclufion is drawn from fome principle, of which we are certain by intuition. With refpect to the firft, we have no fuch knowledge of the nature or inherent properties of any being, material or immaterial, as to draw conclufions from it with certainty. I except not even figure confidered fidered as a quality of matter, though it is the object of mathematical reafoning. As we have no ftandard for determining with precision 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 demonftrative? This question may appear at firft 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? 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 form images of things that never existed; 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 crookednefs, and it is eafy to form an idea of a figure bounded by fuch lines. Such ideal figures are the fubjects of mathematical reafoning; and thefe being perfectly clear and diftinct, are proper fubjects for demonftrative reafoning of the first kind. Mathematical reafoning however is not merely a mental entertainment: it is of real ufe in life, by directing us to operate upon 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 to have nearly the properties of a perfect globe. In a word, though 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 its ideal per fection, with the greater accuracy will the mathematical truth be applicable. The component parts of figures, viz. lines and angles, are extremely 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 straight line; and the child will ufe thefe terms familiarly, without hazard of a mistake. Draw a perpendicular upon paper: let the child advert, that the upward line leans neither to the right nor to the left, and for that reafon is termed a perpendicular: the child will apply that term familiarly to a tree, to the wall of a house, 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 accustomed are we to definitions, that even thefe fimple ideas are not fuffered to efcape. A ftraight line, for example, is defined to be the fhortest that can be drawn between two given points. Is it fo, that even a man, not to talk of a child, can have no idea of a straight line till he be told that the fhorteft line between two points is a ftraight line? How many talk familiarly of a ftraight line who never happened to think of that fact, which is an inference only, not a definition. If I had not beforehand an idea of a ftraight line, I fhould never be able to find out, that it is the fhortest that can be drawn between two points. D'Alembert ftrains hard, but without fuccefs, for a definition of a ftraight line, and of the others mentioned. It is difficult to avoid fmiling at his definition of parallel lines. Draw, fays he, a ftraight line: erect upon it two perpendiculars of the fame length: upon |