J

becaufe from experiments or obfervations already made, you know that if you have two straight strips of wood, for inftance, and put the end of one exactly over the end of the other, and turn the upper ftrip in the direction of the lower, the other end will neither over-reach nor fall fhort of the other end of the lower, but lie exactly over it and this is what we mean by equality of length, a term folely derived from fuch experiments of menfuration, made in early life.

To fhew by experiment, likewife, which alone can fhew it, what an angle is, and moreover that, if the line Ac ftretches as much from AB as DE does from DF, AC must lie along DF, whenever AB lies along DE, a carpenter's ruler may be opened and fhut to various angles, and one carpenter's ruler may be placed over another.

This

This experiment having been performed till the refult is allowed and understood, it will be seen that the point c must fall upon F, from the equality of the lines AC and

DF.

"and

Then BC muft fall upon EF; why?"-make the experiment; you cannot place two straight sticks, or trace two

as to

ftraight lines, fo encompass a space, try as long as you please and fatisfy yourfelf: if you were to try to inclose a piece of land by two straight hedges, you would find all your attempts vain; there would be one or two open ends, place the hedges how you will; and then your field would not be enclosed, but open. Therefore BC must needs confent to fall along EF; otherwife B being upon E, and c upon F, as I have already fhewn you, BC must go on one fide EF like one of the dotted lines; but then BC and EF, two ftraight lines, would

C 4

would enclose a space, which the whole

courfe of your experiments has fhewn they cannot poffibly do.

you,

Then the triangles must exactly fall upon one another, &c.

I have been purposely prolix in this demonstration, to fhew how it begins in experiment, goes on by experiment, and ends in an experimental conclufion. There may be another use in infifting so particularly upon the nature of the reasoning procefs here among those who teach mathematics, without understanding their practical application, and also without entertaining a juft idea of the nature of demonstration, there prevails a fort of pedantry productive of infinite difguft to the learner. If by detached figures I could fhew the truth of any propofition in an inftant, I am forbidden, because this is an unmathematical mode of proceeding; that

is,

is, mathematical reasoning is supposed to be fomething independent of experience, and the science to be more refined than the experimental sciences. Hence, if a Greek writer happens to have written a demonstration a mile long, which demonstration can be nothing but a concatenation of the refults of observation and experiment, I muft take this tedious round, rather than be allowed to arrive at the point desired by only traverfing half a dozen yards, provided this fhorter road leads through the unhallowed region of the fenfes.

The fifth propofition is faid to have stopped many students of geometry in their career; this is owing partly to the length of the demonftration, and partly to the complication of the diagram. The demonstration is, however, nothing but the refult of the experiments in that of the fourth, combined with the refult of two

6

other

other very fimple experiments; of which the one, that if you take equal parts from equal lines, the remainders fhall be equal, will be easily granted from diftinct recollection. The other, that if from equal angles you take equal parts, the remaining angels will be equal, fhould be shewn by two pair of compaffes, or two carpenter's rules opened equally, and then brought nearer together in both an equal and unequal degree. The reason why it becomes neceffary to take pains to make beginners comprehend the nature of an angle, is because in life we do not pay attention to the different expansions of lines meeting at a point. On the contrary, there is not a child but what is accustomed to measure fimple lengths.

I would rather choose to appeal to thefe two experiments, than to the third axiom placed before Euclid's elements, viz. that

if