1 knowledge acquired by means of that perception, not only knowledge in its proper fenfe, but alfo opinion and belief, are termed intuitive knowledge. But there are many things, the knowledge of which is not obtained 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 procefs or operation muft be explained, in order to understand the nature of reafoning. 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, 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 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 process is termed a con- . clufion, being the laft or concluding perception. No fort of reasoning affords fo clear a notion of the foregoing progrefs, as that which is mathematical.Equality is the only mathematical relation; and comparison therefore is the only means by which mathematical propofitions are ascertained. To that science belong a fet of intuitive propofitions, termed axioms, which are all founded on 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, Book III. 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 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 respect. I proceed to fhew 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, which appear by intuition to be equal to both. If this property cannot be difcovered in any one fet of angles, we must go more leifurely 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 angles now difcovered: and fo on in the comparison, till at laft we difcover a fet of angles, equal not only to those thus 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 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 third. I proceed to a different example, which concerns the relation between caufe and effect. The propofition to be demonstrated is, "That there exifts a good and in"telligent Being, who is the cause of all the wife and "benevolent effects that are produced in the govern"ment 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 thefe effects, I begin with an intuitive propofition mentioned above, "That every effect adapted to a good end or purpofe, proceeds from a defigning and benevolent/ "caufe." The next step is, to examine whether man can be the cause he is provided indeed with fome fhare of wisdom and benevolence; but the effects inentioned are far above his power, and no lefs above his wisdom. : Neither 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 invisible being, endowed. with boundless power, goodness, and intelligence; and that invifible being is termed Go D. Reafoning requires two mental powers, namely, the powers of invention, and of perceiving relations. By the former are difcovered 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 reafoning, 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 terined. Hence reafoning is distinguished into two kinds; demonftrative, and probable. Demonftrative reafoning is alfo of two kinds; in the first, 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 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 as a quality of matter, tho' it is the object of mathematical reafoning. As we have no ftandard 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 reasoning to be demonftrative? This queftion may appear at first fight puzzling; and I know not that it has any where been diftinctly explained. Perhaps what follow may be fatisfactory. The fubjects of arithmetical reafoning are numbers. The fubjects of mathematical reafoning are figures.But what figures are fubjects of mathematical reafoning? 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 VOL. III. E a river a river running upward. This power operates upon figures. There is perhaps no figure existing the fides of which are straight lines. But it is easy to form an idea of a line, that has no waving or crookedness in it; 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 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 fpeaking, the fubject 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 extremely imple, 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 ftraight line; and the child will use these 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 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 in applying the fame term to the fides of a door or of a window. Yet fo accustomed are we to definitions, that even these fimple ideas are not fuffered to efcape. A ftraight line, for example, is defined to be the shortest that can be drawn between two given points. The fact 99 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 shortest between two given points. D'Alembert ftrains hard, but without fuccefs, for a definition of a straight line, and of the others mentioned. It is difficult to avoid smiling at his definition of parallel lines. Draw, fays he, a ftraight line : erect upon it two perpendiculars of the fame length: upon their two extremities draw another straight line; and that line is faid to be parallel to the first mentioned : as if, to understand what is meant by the expreffion izvo parallel lines, we must first understand what is meant by a ftraight line, by a perpendicular, and by two lines equal in length. A very flight reflection upon the operations of his own mind, would have taught this author, that he could form the idea of parallel lines without running through so many intermediate fteps: fight alone is fufficient to explain the term to a boy, and even to a girl. At any rate, where is the neceffity of introducing the line last mentioned? If the idea of parallels cannot be obtained from the two perpendiculars alone, the additional line drawn through their extremities will certainly not make it more clear. Mathematical figures being in their nature complex, are capable of being defined; and from the foregoing fimple ideas, it is eafy to define every one of them.For example, a circle is a figure having a point within it, named the centre, through which all the ftraight lines that can be drawn, and extended to the circumference, are equal; a furface bounded by four equal ftraight lines, and having four right angles, is termed a Square; and a cube is a folid, of which all the fix furfaces are fquares. In the investigation of mathematical truths, we affift the imagination, by drawing figures upon paper that refemble our ideas. There is no neceffity for a perfect refemblance: a black spot, which in reality is a small round furface, ferves to reprefent a mathematical point; and a black line, which in reality is a long narrow furface, ferves to represent a mathematical line. When E 2 we |