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vered; and fo on in the comparison, till at laft we difcover a fet of angles, equal not only to thofe 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 thefe 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 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 example the fundamental propofition; which is taken for granted, because it is verified by experience. In order to discover the caufe of these 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." Bb 2

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The next step is, to examine whether man can be the caufe: he is provided indeed with fome fhare of wifdom 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 stars; 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 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 proposition and to the conclufion: 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 termed. Henee reafoning is diftinguifhed into two kinds; demonftrative, and probable. Demon

ftrative

ftrative reafoning 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

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 follows may be fatisfactory.

The fubjects of arithmetical reafoning are numbers. The fubjects of mathematical reasoning are figures. But what figures? Not fuch as I fee; but fuch as I

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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 straight lines'; but it is eafy to form an idea of a line that has no waving or crookedness, and it is easy 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 demonstrative reasoning of the first kind. Mathematical reasoning 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, tho' ideas are, properly fpeaking, the fubject of mathematical evidence; yet the end and purpofe of that evidence is, to direct us with refpect to figures as they really exift; and

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the nearer any real figure approaches to its ideal perfection, 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 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 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 thefe fimple ideas are not fuffered to escape. A traight line, for example, is defined to be

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