than we meet with them at every step in every demonstration; and I fhall, I hope, be allowed to have established it firmly, if I fhew that Euclid fets out from experiments, and proceeds onwards by their aid, appealing conftantly to what we have already learned from the exercife of our fenfes, or may immediately learn. The fame thing must needs be equally true of every other elementary author. After having exemplified the nature of demonftrative reasoning, I fhall leave the reader to extend this mode of confidering it to other cafes, in a full perfuafion that he will find the fame process repeated, in every demonftration upon which he may choose to make the trial. In order therefore to avoid needlefs repetition, I fhall pass on, and, fecondly, apply the principle to the folution of certain difficulties in the elements of geometry, about which a great deal has

been written. It will appear, that neither the doctrine of parallel lines, nor the fifth definition of the fifth book, on which Euclid founds his reasoning concerning proportion, have any thing obfcure or doubtful, when properly confidered. Lastly, to take off that glofs of novelty, which fo much fcandalizes mankind, when truth appears before their eyes for the first time, I fhall fhew that Mr. Locke has diftinctly announced the fame opinion. The reader may therefore, if he chooses, confider what follows as a commentary, upon a paffage of Mr. Locke, which was totally loft upon me, and, as it appears, upon others. A good commentary would prevent or relieve much perplexity.

I am first to review the outfet of Euclid's reasoning, in order to fhew that he begins from experiments, and proceeds by experiments. These experiments may, indeed,

indeed, be called mental experiments, fince the appeal is made to recollection, and they are commonly repeated in thought. But they are not at all more independant of experience than my acquired knowledge of the refult of an experiment with a balance of which one arm fhould be ten inches, and the other one inch long, and each arm should be loaded in an inverse proportion to its length: or, to take another example, fuppofe a train of reasoning were to set out on the supposition of iron yielding hydrogene air, while it is diffolving in vitriolic acid of a certain ftrength, I should most affuredly not think it neceffary to repeat the experiment. Now juft as familiar as fuch refult is to a person at all converfant in chemistry, just so familiar to every person, at least every one arrived at the years of discretion, and having the use of his hands and eyes, are many of the experiments of geometry.

a

The fourth propofition of the first book is the first theorem occurring in the elements of Euclid; this propofition, and the axiom as it is called, upon which the demonftration is founded, viz. the 10th in Simfon's edition, may be regarded as the corner-stone of geometrical reasoning. The proposition is this:

If two triangles have two fides of the one equal to two fides of the other each to each and have likewife the angles contained by thofe fides equal to one another, they shall have their bafes, or third fides, equal; and the two triangles fhall be equal; and their other angles fhall be equal, each to each, viz. those to which the equal fides are oppofite.

A A

B

It

It will be granted, that the ideas of a triangle, of equal or unequal expansions between lines meeting at a point, and of the equal or unequal length of different ftraight lines, are all acquired by the exercife of the fenfes. It is, indeed, evident, that we cannot in

quire them.

any other

way ac

Now the fide AB being made equal to DE, and AC to DF, and the angles BAC and EDF being equal, let us obferve the experimental process of the demonstration.

If you have a model of each triangle cut out in pasteboard, or any other material, you are to place the point A upon D, which is to commence an experiment; if the triangles be only traced upon a furface, you are to imagine A placed upon D, which is to imagine the commencement of an experiment. Then you are to lay AB along DE; now в will fall upon E. B

C 3

66

Why?" because