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the shortest 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 ftraight line till he be told that the fhortest 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 fhorteft 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: crect upon it two perpendiculars of the fame length: upon their two extremities draw another ftraight 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 muft 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 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 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 straight lines, and having four right angles, is termed a fquare; 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:
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 reprefent a mathematical line. When we reafon about the figures compofed of fuch lines, it is fufficient that these figures have fome appearance of regularity lefs or more is of no importance; because our reafoning 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. After tracing the steps that lead to the conclufion, we are fatisfied that the propofition is true; being confcious that the reasoning is built upon the ideal figure, not upon that which is drawn upon the paper. And being alfo confcious, 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 whatever.
Numbers confidered by themselves, abftractedly from things, make the fubject
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 fingle dafh of the pen, exprefs clearly what would require many words. By that means, a very long chain of reasoning 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 multitude, 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 reprefent 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 say, 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 the reafoning confifts of many fteps. And tho' on a reCc 2 view
view no error be discovered, yet he is confcious that there may be errors, tho' they have escaped him.
As to the other kind of demonstrative reafoning, founded on propofitions of which we are intuitively certain; I juftly call it demonftrative, because it affords the fame conviction that arifes from mathematical reasoning. In both, the means of conviction are the fame, viz. a clear perception of the relation between two ideas : and there are many relations of which we have ideas no less clear than of equality; witness fubftance and quality, the whole and its parts, caufe and effect, and many others. From the intuitive propofition, for example, That nothing which begins to exift can exift without a caufe, I can conclude, that some one being must have existed from all eternity, with no less certainty, than that the three angles of a triangle are equal to two right angles.
What falls next in order, is that inferior fort of knowledge which is termed opinion; and which, like knowledge properly fo termed, is founded in fome inftances upon intuition, and in fome upon reafoning. But it differs from knowledge