angles now difcovered; and fo on in the comparison, til at last we discover a fet of angles, equal not only to thofe thus introduced, but also 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 respect, is equal to a third.

I proceed to a different example, which concerns the relation between caufe and effect. The propofition to be demonftrated is, "That there exifts 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, I begin with an intuitive propofition mentioned above, "That every effect adapted to a good end

or purpose, proceeds from a designing and benevolent caufe." The next step is, to examine whether man can be the caufe: he is provided indeed with some share of wisdom and benevolence; but the effects mentioned are far above his power, and no less above his wifdom. Neither can this earth be the caufe, nor the fun, the moon, the stars; 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 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 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 distinguished into two kinds; demonstrative, and probable. Demonstrative reasoning is also 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 respect to the first, 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 standard 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 reasoning to be demonftrative? This queftion may appear at first sight puzzling; and I know not that it has any where been distinctly explained. Perhaps what follows may be fatisfactory.

The fubjects of arithmetical reasoning are numbers. The subjects of mathematical reasoning are figures. But what figures are fubjects of mathematical reasoning? Not fuch as I fee; but fuch as I form an idea of, abstracting from every imperfection. I explain myself. There is a power in man to form 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 crookedness in it; and it is easy to form an idea of a figure bounded by fuch lines. Such ideal figures are the fubjects of mathematical reasoning; and these being perfectly clear and distinct, are proper fubjects for demonstrative 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 demonstrated 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 respect 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 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 ftraight 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 reafon is termed a perpendicular: the child will apply that term familiarily 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 these simple ideas are not fuffered to escape. A ftraight line, for example, is defined to be the shortest 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 straight line: and had I not beforehand a clear idea of a ftraight line, I could not infer that it is the fhorteft between two given points. D'Alembert

D'Alembert strains hard, but without fuccefs, for a definition of a ftraight 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 two parallel lines, we must first understand what is meant by a straight 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 steps: 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 laft mentioned? If the idea of parallels cannot be obtained from the two perpendiculars alone, the additional line drawn through their extrémities 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 easy 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 straight lines, and having four right angles, is termed a Square; and a cube is a folid, of which all the fix furfaces are squares.

In the investigation of mathematical truths, we affist the imagination, by drawing figures upon paper that resemble our ideas. There is no neceflity for a perfect resemblance: a black spot, which in reality is a finall round furface, ferves to represent a mathematical point; and a black line, which in reality is a long narrow furface, ferves to represent a mathematical line. When we reason about the figures compofed of fuch lines, it is fufficient

that

that thefe figures have fome appearance of regularity: lefs or more is of no importance; because our reasoning is not founded upon them, but upon our ideas. Thus, to demonftrate that the three angles of a triangle are equal to two right angles, a triangle is drawn upon paper, in order to keep the mind steady to its object, and to prevent wandering. After tracing the steps that lead to the conclufion, we are fatisfied that the propofition is true; being conscious that the reafoning is built upon the ideal figure, not upon that which is drawn upon the paper. And being alfo conscious that the enquiry is carried on independent of any particular length of the fides, we are fatisfied of the univerfality of the propofition, and of its being applicable to all triangles what

ever.

Numbers confidered by themfelves, abftractedly from things, make the subject of arithmetic. And with refpect both to mathematical and arithmetical reasonings, which frequently confift of many steps, the process is fhortened by the invention of figns, which, by a single dafh of the pen, exprefs clearly what would require many words. By that means, a very long chain of reafoning is expreffed by a few fymbols; a method that contributes greatly to readiness of comprehenfion. If in fuch reasonings words were neceffary, the mind, embarraffed with their multiplicity, would have great difficulty to follow any long chain of reafoning. A line drawn upon paper reprefents an ideal line, and a few fimple characters represent the abstract ideas of number.

Arithmetical reasoning, like mathematical, depends entirely upon the relation of equality, which can be afcertained with the greatest certainty among many ideas. Hence, reafonings upon fuch ideas afford the highest degree of conviction. I do not fay, however, that this is always the cafe; for a man who is conscious of his own fallibility, is feldom without fome degree of diffidence,

where