the nearer any real figure approaches to its ideal perfection, with the greater accuwill the mathematical truth be appli

racy cable.

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 ufe thefe 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 accustomed are we to definitions, that even these fimple ideas are not suffered to escape. A traight line, for example, is defined to be

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the

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 straight line till he be told that the shortest line between two points is a ftraight line? How many talk familiarly of a straight 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 straight line, I should never be able to find out, that it is the shortest that can be drawn between two 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 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 fo 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 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 straight lines that can be drawn, and extended to the circumference, are equal; a surface 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 assist the imagination, by drawing figures upon paper that resemble our ideas. There is no neceffity for a perfect refemblance:

refemblance: a black fpot, which in reality is a finall round surface, ferves to represent 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 thefe 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 demonstrate 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

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 dash of the pen, express 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 reafonings 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 represents 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 fay, however, that this is always the cafe; for a man who is confcious of his own fallibility, is feldom without fome degree of diffidence, where the reasoning confifts of many steps. And tho' on a re