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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 single part of the other. Second: Take ten of these 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 equal; or add equal parts, and the fums will be equal. Third: If two things be, in the fame respect, equal to a third, the one is equal to the other in the fame refpect. I proceed to fhow 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 demonstrate 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 leisurely to work, and try to find angles that are equal to the three angles of a triangle. These being discovered, we next try to find other angles equal to the angles now difco

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vered; and fo on in the comparison, till at last we discover 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.

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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 discover the cause of these effects, I begin with an intuitive proposition mentioned above, That every effect adapted "to a good end or purpose, proceeds " from a defigning and benevolent cause." The

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 wisdom. 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 thefe be excluded, we are unavoidably led to an invifible being, endowed with boundlefs power, goodnefs, and intelligence; and that invisible 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; demonstrative, and probable. Demon

ftrative

strative 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 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 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 demonftrative? This queftion may appear at first fight 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? Not fuch as I fee; but fuch as I VOL. III, Ce

form

form an idea of, abftracting 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 easy 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 these 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 use in life, by directing us to operate upon matter. There poffibly may not be found any where a 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 speaking, the fubject of mathehatical 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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